Fuzzy logic h l m From Wikipedia, the free encyclopedia Contents
1 Adaptive neuro fuzzy inference system 1 1.1 References ...... 1
2 Bate’s chip 2 2.1 References ...... 2
3 BL (logic) 3 3.1 Syntax ...... 3 3.1.1 Language ...... 3 3.1.2 Axioms ...... 4 3.2 Semantics ...... 4 3.3 Bibliography ...... 5 3.4 References ...... 5
4 Combs method 6 4.1 Equality proof ...... 6 4.2 Combinatorial explosion ...... 6 4.3 Example ...... 6 4.4 References ...... 7
5 Construction of t-norms 8 5.1 Generators of t-norms ...... 8 5.1.1 Additive generators ...... 8 5.1.2 Multiplicative generators ...... 9 5.2 Parametric classes of t-norms ...... 9 5.2.1 Schweizer–Sklar t-norms ...... 10 5.2.2 Hamacher t-norms ...... 10 5.2.3 Frank t-norms ...... 11 5.2.4 Yager t-norms ...... 11 5.2.5 Aczél–Alsina t-norms ...... 11 5.2.6 Dombi t-norms ...... 12 5.2.7 Sugeno–Weber t-norms ...... 12 5.3 Ordinal sums ...... 13 5.3.1 Ordinal sums of continuous t-norms ...... 14
i ii CONTENTS
5.4 Rotations ...... 14 5.5 See also ...... 15 5.6 References ...... 16
6 Defuzzification 17 6.1 Methods ...... 17 6.2 Notes ...... 18 6.3 See also ...... 18
7 Degree of truth 19 7.1 See also ...... 19 7.2 Bibliography ...... 19
8 European Society for Fuzzy Logic and Technology 20 8.1 History ...... 20 8.2 Conferences ...... 20 8.3 Publications ...... 21 8.4 Presidents ...... 21 8.5 References ...... 21 8.6 External links ...... 21
9 Fuzzy architectural spatial analysis 22 9.1 Overview ...... 22 9.2 References ...... 22 9.3 Further reading ...... 22 9.4 See also ...... 23
10 Fuzzy associative matrix 24
11 Fuzzy classification 25 11.1 Classification ...... 25 11.2 See also ...... 26 11.3 References ...... 26
12 Fuzzy cognitive map 27 12.1 Details ...... 27 12.2 References ...... 29 12.3 External links ...... 29
13 Fuzzy Control Language 30 13.1 External links ...... 30
14 Fuzzy control system 31 14.1 Overview ...... 31 14.2 History and applications ...... 31 CONTENTS iii
14.3 Fuzzy sets ...... 32 14.3.1 Fuzzy control in detail ...... 33 14.3.2 Building a fuzzy controller ...... 37 14.4 Antilock brakes ...... 38 14.5 Logical interpretation of fuzzy control ...... 39 14.6 See also ...... 39 14.7 References ...... 40 14.8 Further reading ...... 40 14.9 External links ...... 40
15 Fuzzy electronics 41 15.1 See also ...... 41 15.2 Bibliography ...... 41 15.3 External links ...... 41
16 Fuzzy finite element 42 16.1 See also ...... 42 16.2 References ...... 42
17 Fuzzy logic 43 17.1 Overview ...... 43 17.1.1 Applying truth values ...... 43 17.1.2 Linguistic variables ...... 44 17.2 Early applications ...... 44 17.3 Example ...... 44 17.3.1 Hard science with IF-THEN rules ...... 44 17.3.2 Define with multiply ...... 45 17.3.3 Define with sigmoid ...... 45 17.4 Logical analysis ...... 45 17.4.1 Propositional fuzzy logics ...... 45 17.4.2 Predicate fuzzy logics ...... 45 17.4.3 Decidability issues for fuzzy logic ...... 46 17.5 Fuzzy databases ...... 46 17.6 Comparison to probability ...... 46 17.7 Relation to ecorithms ...... 47 17.8 Compensatory fuzzy logic ...... 47 17.9 See also ...... 47 17.10References ...... 48 17.11Bibliography ...... 49 17.12External links ...... 51
18 Fuzzy markup language 52 18.1 Overview ...... 52 iv CONTENTS
18.2 FML at work: syntax, grammar and hardware synthesis ...... 52 18.2.1 FML Syntax ...... 53 18.2.2 FML Grammar ...... 55 18.2.3 FML Synthesis ...... 56 18.3 References ...... 56 18.4 Further reading ...... 57
19 Fuzzy mathematics 58 19.1 Some fields of mathematics using fuzzy set theory ...... 58 19.2 See also ...... 59 19.3 References ...... 59 19.4 External links ...... 60
20 Fuzzy measure theory 61 20.1 Definitions ...... 61 20.2 Properties of fuzzy measures ...... 61 20.3 Möbius representation ...... 62 20.4 Simplification assumptions for fuzzy measures ...... 62 20.4.1 Sugeno λ-measure ...... 62 20.4.2 k-additive fuzzy measure ...... 63 20.5 Shapley and interaction indices ...... 63 20.6 See also ...... 63 20.7 References ...... 63 20.8 External links ...... 63
21 Fuzzy number 64 21.1 See also ...... 64 21.2 References ...... 64 21.3 External links ...... 64 21.4 Applications ...... 64
22 Fuzzy pay-off method for real option valuation 65 22.1 Method ...... 65 22.2 Use of the method ...... 66 22.3 References ...... 66 22.4 External links ...... 66
23 Fuzzy routing 67 23.1 See also ...... 67 23.2 External links ...... 67
24 Fuzzy rule 68 24.1 Comparison between Boolean and fuzzy logic rules ...... 68 24.2 Comparison between computational verb and fuzzy logic rules ...... 68 CONTENTS v
24.3 See also ...... 68
25 Fuzzy set 69 25.1 Definition ...... 69 25.2 Fuzzy logic ...... 69 25.3 Fuzzy number ...... 70 25.4 Fuzzy interval ...... 70 25.5 Fuzzy relation equation ...... 70 25.6 Axiomatic definition of credibility ...... 70 25.7 Credibility inversion theorem ...... 70 25.8 Expected Value ...... 71 25.9 Entropy ...... 71 25.10Generalizations ...... 71 25.11See also ...... 72 25.12References ...... 73 25.13Further reading ...... 73 25.14External links ...... 75
26 Fuzzy set operations 76 26.1 Standard fuzzy set operations ...... 76 26.2 Fuzzy complements ...... 76 26.2.1 Axioms for fuzzy complements ...... 77 26.3 Fuzzy intersections ...... 77 26.3.1 Axioms for fuzzy intersection ...... 77 26.4 Fuzzy unions ...... 77 26.4.1 Axioms for fuzzy union ...... 78 26.5 Aggregation operations ...... 78 26.5.1 Axioms for aggregation operations fuzzy sets ...... 78 26.6 See also ...... 78 26.7 Further reading ...... 78 26.8 External References ...... 78
27 Fuzzy Sets and Systems 79 27.1 See also ...... 79
28 Fuzzy subalgebra 80 28.1 Definition ...... 80 28.2 Fuzzy subgroups and submonoids ...... 80 28.3 Bibliography ...... 81
29 Fuzzy transportation 82
30 High Performance Fuzzy Computing 83 30.1 External links ...... 83 vi CONTENTS
31 Linear partial information 84 31.1 Definition ...... 84 31.2 Applications ...... 84 31.3 Fuzzy equilibrium and stability ...... 84 31.4 LPI equilibrium point ...... 85 31.5 See also ...... 85 31.6 Selected references ...... 85 31.7 External links ...... 85
32 Membership function (mathematics) 87 32.1 Definition ...... 87 32.2 Capacity ...... 88 32.3 See also ...... 88 32.4 References ...... 88 32.5 Bibliography ...... 88 32.6 External links ...... 88
33 Monoidal t-norm logic 89 33.1 Motivation ...... 89 33.2 Syntax ...... 89 33.2.1 Language ...... 90 33.2.2 Axioms ...... 91 33.3 Semantics ...... 91 33.3.1 General semantics ...... 91 33.3.2 Linear semantics ...... 93 33.3.3 Standard semantics ...... 93 33.4 Bibliography ...... 93 33.5 References ...... 94
34 MV-algebra 95 34.1 Definitions ...... 95 34.2 Examples of MV-algebras ...... 95 34.3 Relation to Łukasiewicz logic ...... 96 34.3.1 MVn-algebras ...... 96 34.4 Relation to functional analysis ...... 97 34.5 In software ...... 97 34.6 References ...... 97 34.7 Further reading ...... 97 34.8 External links ...... 98
35 Noise-based logic 99 35.1 The noise-based logic space and hyperspace ...... 99 35.2 The types of signals used in noise-based logic ...... 99 CONTENTS vii
35.3 The noise-based logic gates ...... 99 35.4 Universality of noise-based logic ...... 100 35.5 Computation by noise-based logic ...... 100 35.6 Computer chip realization of noise-based logic ...... 100 35.7 References ...... 100 35.8 External links ...... 100
36 SQLf 101 36.1 Basic Block ...... 101 36.2 References ...... 102
37 Łukasiewicz logic 103 37.1 Language ...... 103 37.2 Axioms ...... 103 37.3 Real-valued semantics ...... 104 37.4 Finite-valued and countable-valued semantics ...... 104 37.5 General algebraic semantics ...... 104 37.6 References ...... 105 37.7 Further reading ...... 105 37.8 Text and image sources, contributors, and licenses ...... 106 37.8.1 Text ...... 106 37.8.2 Images ...... 108 37.8.3 Content license ...... 110 Chapter 1
Adaptive neuro fuzzy inference system
An adaptive neuro-fuzzy inference system or adaptive network-based fuzzy inference system (ANFIS) is a kind of artificial neural network that is based on Takagi–Sugeno fuzzy inference system. The technique was developed in the early 1990s.[1][2] Since it integrates both neural networks and fuzzy logic principles, it has potential to capture the benefits of both in a single framework. Its inference system corresponds to a set of fuzzy IF–THEN rules that have learning capability to approximate nonlinear functions.[3] Hence, ANFIS is considered to be a universal estimator.[4]
1.1 References
[1] Jang, Jyh-Shing R (1991). Fuzzy Modeling Using Generalized Neural Networks and Kalman Filter Algorithm (PDF). Pro- ceedings of the 9th National Conference on Artificial Intelligence, Anaheim, CA, USA, July 14–19 2. pp. 762–767.
[2] Jang, J.-S.R. (1993). “ANFIS: adaptive-network-based fuzzy inference system”. IEEE Transactions on Systems, Man and Cybernetics 23 (3). doi:10.1109/21.256541.
[3] Abraham, A. (2005), “Adaptation of Fuzzy Inference System Using Neural Learning”, in Nedjah, Nadia; de Macedo Mourelle, Luiza, Fuzzy Systems Engineering: Theory and Practice, Studies in Fuzziness and Soft Computing 181, Germany: Springer Verlag, pp. 53–83, doi:10.1007/11339366_3
[4] Jang, Sun, Mizutani (1997) – Neuro-Fuzzy and Soft Computing – Prentice Hall, pp 335–368, ISBN 0-13-261066-3
1 Chapter 2
Bate’s chip
Bates’s chip (also called a sloppy chip or fuzzy chip) is a theoretical chip proposed by MIT Media Lab's computer scientist Joseph Bates that would incorporate fuzzy logic to do calculations. The resulting calculations would be less accurate, though they would be performed significantly faster.[1][2]
2.1 References
[1] Bennett, Drake (January 28, 2011). “Innovator: Joseph Bates - BusinessWeek”. Bloomberg Businessweek. Retrieved 30 January 2011.
[2] Hardesty, Larry (January 3, 2011). “The surprising usefulness of sloppy arithmetic”. MIT News Office. Archived from the original on 29 January 2011. Retrieved 30 January 2011.
2 Chapter 3
BL (logic)
Basic fuzzy Logic (or shortly BL), the logic of continuous t-norms, is one of t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices;[1] it extends the logic of all left-continuous t-norms MTL.
3.1 Syntax
3.1.1 Language
The language of the propositional logic BL consists of countably many propositional variables and the following primitive logical connectives:
• Implication → (binary) • Strong conjunction ⊗ (binary). The sign & is a more traditional notation for strong conjunction in the literature on fuzzy logic, while the notation ⊗ follows the tradition of substructural logics. • Bottom ⊥ (nullary — a propositional constant); 0 or 0 are common alternative signs and zero a common alternative name for the propositional constant (as the constants bottom and zero of substructural logics coincide in MTL).
The following are the most common defined logical connectives:
• Weak conjunction ∧ (binary), also called lattice conjunction (as it is always realized by the lattice operation of meet in algebraic semantics). Unlike MTL and weaker substructural logics, weak conjunction is definable in BL as
A ∧ B ≡ A ⊗ (A → B)
• Negation ¬ (unary), defined as
¬A ≡ A → ⊥
• Equivalence ↔ (binary), defined as
A ↔ B ≡ (A → B) ∧ (B → A) As in MTL, the definition is equivalent to (A → B) ⊗ (B → A).
3 4 CHAPTER 3. BL (LOGIC)
• (Weak) disjunction ∨ (binary), also called lattice disjunction (as it is always realized by the lattice operation of join in algebraic semantics), defined as
A ∨ B ≡ ((A → B) → B) ∧ ((B → A) → A)
• Top ⊤ (nullary), also called one and denoted by 1 or 1 (as the constants top and zero of substructural logics coincide in MTL), defined as
⊤ ≡ ⊥ → ⊥
Well-formed formulae of BL are defined as usual in propositional logics. In order to save parentheses, it is common to use the following order of precedence:
• Unary connectives (bind most closely) • Binary connectives other than implication and equivalence • Implication and equivalence (bind most loosely)
3.1.2 Axioms
A Hilbert-style deduction system for BL has been introduced by Petr Hájek (1998). Its single derivation rule is modus ponens:
from A and A → B derive B.
The following are its axiom schemata:
(BL1):(A → B) → ((B → C) → (A → C)) (BL2): A ⊗ B → A (BL3): A ⊗ B → B ⊗ A (BL4): A ⊗ (A → B) → B ⊗ (B → A) (BL5a):(A → (B → C)) → (A ⊗ B → C) (BL5b):(A ⊗ B → C) → (A → (B → C)) (BL6): ((A → B) → C) → (((B → A) → C) → C) (BL7): ⊥ → A
The axioms (BL2) and (BL3) of the original axiomatic system were shown to be redundant (Chvalovský, 2012) and (Cintula, 2005). All the other axioms were shown to be independent (Chvalovský, 2012).
3.2 Semantics
Like in other propositional t-norm fuzzy logics, algebraic semantics is predominantly used for BL, with three main classes of algebras with respect to which the logic is complete:
• General semantics, formed of all BL-algebras — that is, all algebras for which the logic is sound • Linear semantics, formed of all linear BL-algebras — that is, all BL-algebras whose lattice order is linear • Standard semantics, formed of all standard BL-algebras — that is, all BL-algebras whose lattice reduct is the real unit interval [0, 1] with the usual order; they are uniquely determined by the function that interprets strong conjunction, which can be any continuous t-norm 3.3. BIBLIOGRAPHY 5
3.3 Bibliography
• Hájek P., 1998, Metamathematics of Fuzzy Logic. Dordrecht: Kluwer.
• Ono, H., 2003, “Substructural logics and residuated lattices — an introduction”. In F.V. Hendricks, J. Mali- nowski (eds.): Trends in Logic: 50 Years of Studia Logica, Trends in Logic 20: 177–212.
• Cintula P., 2005, “Short note: On the redundancy of axiom (A3) in BL and MTL”. Soft Computing 9: 942.
• Chvalovský K., 2012, "On the Independence of Axioms in BL and MTL". Fuzzy Sets and Systems 197: 123– 129, doi:10.1016/j.fss.2011.10.018.
3.4 References
[1] Ono (2003). Chapter 4
Combs method
The Combs method is a method of writing fuzzy logic rules described by William E. Combs in 1997. It is designed to prevent combinatorial explosion in fuzzy logic rules. The Combs method takes advantage of the logical equality ((p ∧ q) ⇒ r) ⇐⇒ ((p ⇒ r) ∨ (q ⇒ r)) .
4.1 Equality proof
The simplest proof of given equality involves usage of truth tables:
4.2 Combinatorial explosion
Suppose we have a fuzzy system that considers N variables at a time, each of which can fit into at least one of S sets. The number of rules necessary to cover all the cases in a traditional fuzzy system is SN , whereas the Combs method would need only S × N rules. For example, if we have five sets and five variables to consider to produce one output, covering all the cases would require 3125 rules in a traditional system, while the Combs method would require only 25 rules, taming the combinatorial explosion that occurs when more inputs or more sets are added to the system. This article will focus on the Combs method itself. To learn more about the way rules are traditionally formed, see fuzzy logic and fuzzy associative matrix.
4.3 Example
Suppose we were designing an artificial personality system that determined how friendly the personality is supposed to be towards a person in a strategic video game. The personality would consider its own fear, trust, and love in the other person. A set of rules in the Combs system might look like this: The table translates to: [IF Fear IS Unafraid THEN Friendship IS Enemies OR IF Fear IS ModerateFear THEN Friendship IS Neutral OR IF Fear IS Afraid THEN Friendship IS GoodFriends ] OR [IF Trust IS Distrusting THEN Friendship IS Enemies OR IF Trust IS ModerateTrust THEN Friendship IS Neutral OR IF Trust IS Trusting THEN Friendship IS GoodFriends] OR [IF Love IS Unloving THEN Friendship IS Enemies OR IF Love IS ModerateLove THEN Friendship IS Neutral OR IF Love IS Loving THEN Friendship IS GoodFriends] In this case, because the table follows a straightforward pattern in the output, it could be rewritten as: Each column of the table maps to the output provided in the last row. To obtain the output of the system, we just average the outputs of each rule for that output. For example, to calculate how much the computer is Enemies with the player, we take the average of how much the computer is Unafraid, Distrusting, and Unloving of the player. When all three averages are obtained, the result can then be defuzzified by any of the traditional means.
6 4.4. REFERENCES 7
4.4 References
• The Combs Method for Rapid Inference (the original paper by William E. Combs) Chapter 5
Construction of t-norms
In mathematics, t-norms are a special kind of binary operations on the real unit interval [0, 1]. Various constructions of t-norms, either by explicit definition or by transformation from previously known functions, provide a plenitude of examples and classes of t-norms. This is important, e.g., for finding counter-examples or supplying t-norms with particular properties for use in engineering applications of fuzzy logic. The main ways of construction of t-norms include using generators, defining parametric classes of t-norms, rotations, or ordinal sums of t-norms. Relevant background can be found in the article on t-norms.
5.1 Generators of t-norms
The method of constructing t-norms by generators consists in using a unary function (generator) to transform some known binary function (most often, addition or multiplication) into a t-norm. In order to allow using non-bijective generators, which do not have the inverse function, the following notion of pseudo-inverse function is employed:
Let f:[a, b] → [c, d] be a monotone function between two closed subintervals of extended real line. The pseudo-inverse function to f is the function f (−1):[c, d] → [a, b] defined as { sup{x ∈ [a, b] | f(x) < y} forfnon-decreasing f (−1)(y) = sup{x ∈ [a, b] | f(x) > y} forfnon-increasing.
5.1.1 Additive generators
The construction of t-norms by additive generators is based on the following theorem:
Let f: [0, 1] → [0, +∞] be a strictly decreasing function such that f(1) = 0 and f(x) + f(y) is in the range of f or equal to f(0+) or +∞ for all x, y in [0, 1]. Then the function T: [0, 1]2 → [0, 1] defined as T(x, y) = f (−1)(f(x) + f(y)) is a t-norm.
If a t-norm T results from the latter construction by a function f which is right-continuous in 0, then f is called an additive generator of T. Examples:
• The function f(x) = 1 – x for x in [0, 1] is an additive generator of the Łukasiewicz t-norm. • The function f defined as f(x) = –log(x) if 0 < x ≤ 1 and f(0) = +∞ is an additive generator of the product t-norm.
8 5.2. PARAMETRIC CLASSES OF T-NORMS 9
• The function f defined as f(x) = 2 – x if 0 ≤ x < 1 and f(1) = 0 is an additive generator of the drastic t-norm.
Basic properties of additive generators are summarized by the following theorem:
Let f: [0, 1] → [0, +∞] be an additive generator of a t-norm T. Then:
• T is an Archimedean t-norm. • T is continuous if and only if f is continuous. • T is strictly monotone if and only if f(0) = +∞. • Each element of (0, 1) is a nilpotent element of T if and only if f(0) < +∞. • The multiple of f by a positive constant is also an additive generator of T. • T has no non-trivial idempotents. (Consequently, e.g., the minimum t-norm has no additive gen- erator.)
5.1.2 Multiplicative generators
The isomorphism between addition on [0, +∞] and multiplication on [0, 1] by the logarithm and the exponential function allow two-way transformations between additive and multiplicative generators of a t-norm. If f is an additive generator of a t-norm T, then the function h: [0, 1] → [0, 1] defined as h(x) = e−f (x) is a multiplicative generator of T, that is, a function h such that
• h is strictly increasing
• h(1) = 1
• h(x)· h(y) is in the range of h or equal to 0 or h(0+) for all x, y in [0, 1]
• h is right-continuous in 0
• T(x, y) = h (−1)(h(x)· h(y)).
Vice versa, if h is a multiplicative generator of T, then f: [0, 1] → [0, +∞] defined by f(x) = −log(h(x)) is an additive generator of T.
5.2 Parametric classes of t-norms
Many families of related t-norms can be defined by an explicit formula depending on a parameter p. This section lists the best known parameterized families of t-norms. The following definitions will be used in the list:
• A family of t-norms Tp parameterized by p is increasing if Tp(x, y) ≤ Tq(x, y) for all x, y in [0, 1] whenever p ≤ q (similarly for decreasing and strictly increasing or decreasing).
• A family of t-norms Tp is continuous with respect to the parameter p if
lim Tp = Tp0 p→p0
for all values p0 of the parameter. 10 CHAPTER 5. CONSTRUCTION OF T-NORMS
Graph (3D and contours) of the Schweizer–Sklar t-norm with p = 2
5.2.1 Schweizer–Sklar t-norms
The family of Schweizer–Sklar t-norms, introduced by Berthold Schweizer and Abe Sklar in the early 1960s, is given by the parametric definition
−∞ Tmin(x, y) ifp = (xp + yp − 1)1/p if − ∞ < p < 0 SS Tp (x, y) = Tprod(x, y) ifp = 0 (max(0, xp + yp − 1))1/p if0 < p < +∞ TD(x, y) ifp = +∞.
SS A Schweizer–Sklar t-norm Tp is
• Archimedean if and only if p > −∞
• Continuous if and only if p < +∞
• Strict if and only if −∞ < p ≤ 0 (for p = −1 it is the Hamacher product)
• Nilpotent if and only if 0 < p < +∞ (for p = 1 it is the Łukasiewicz t-norm).
The family is strictly decreasing for p ≥ 0 and continuous with respect to p in [−∞, +∞]. An additive generator for SS Tp for −∞ < p < +∞ is
{ − SS log x ifp = 0 fp (x) = 1−xp p otherwise.
5.2.2 Hamacher t-norms
The family of Hamacher t-norms, introduced by Horst Hamacher in the late 1970s, is given by the following para- metric definition for 0 ≤ p ≤ +∞: 5.2. PARAMETRIC CLASSES OF T-NORMS 11
TD(x, y) ifp = +∞ H Tp (x, y) = 0 ifp = x = y = 0 xy p+(1−p)(x+y−xy) otherwise.
H The t-norm T0 is called the Hamacher product. H Hamacher t-norms are the only t-norms which are rational functions. The Hamacher t-norm Tp is strict if and only if p < +∞ (for p = 1 it is the product t-norm). The family is strictly decreasing and continuous with respect to p. An H additive generator of Tp for p < +∞ is
{ 1−x H x ifp = 0 fp (x) = p+(1−p)x log x otherwise.
5.2.3 Frank t-norms
The family of Frank t-norms, introduced by M.J. Frank in the late 1970s, is given by the parametric definition for 0 ≤ p ≤ +∞ as follows:
T (x, y) ifp = 0 min T (x, y) ifp = 1 F prod Tp (x, y) = ∞ TLuk(x, y) ) ifp = + (px−1)(py −1) logp 1 + p−1 otherwise.
F The Frank t-norm Tp is strict if p < +∞. The family is strictly decreasing and continuous with respect to p. An F additive generator for Tp is
− log x ifp = 1 F − ∞ fp (x) = 1 x ifp = + p−1 log px−1 otherwise.
5.2.4 Yager t-norms
The family of Yager t-norms, introduced in the early 1980s by Ronald R. Yager, is given for 0 ≤ p ≤ +∞ by
TD(x,( y) ) ifp = 0 T Y(x, y) = max 0, 1 − ((1 − x)p + (1 − y)p)1/p if0 < p < +∞ p Tmin(x, y) ifp = +∞
Y The Yager t-norm Tp is nilpotent if and only if 0 < p < +∞ (for p = 1 it is the Łukasiewicz t-norm). The family is Y strictly increasing and continuous with respect to p. The Yager t-norm Tp for 0 < p < +∞ arises from the Łukasiewicz Y t-norm by raising its additive generator to the power of p. An additive generator of Tp for 0 < p < +∞ is
Y − p fp (x) = (1 x) .
5.2.5 Aczél–Alsina t-norms
The family of Aczél–Alsina t-norms, introduced in the early 1980s by János Aczél and Claudi Alsina, is given for 0 ≤ p ≤ +∞ by 12 CHAPTER 5. CONSTRUCTION OF T-NORMS
Graph of the Yager t-norm with p = 2
TD(x, y) ifp = 0 p p 1/p T AA(x, y) = e−(| log x| +| log y| ) if0 < p < +∞ p Tmin(x, y) ifp = +∞
AA The Aczél–Alsina t-norm Tp is strict if and only if 0 < p < +∞ (for p = 1 it is the product t-norm). The family is AA strictly increasing and continuous with respect to p. The Aczél–Alsina t-norm Tp for 0 < p < +∞ arises from the AA product t-norm by raising its additive generator to the power of p. An additive generator of Tp for 0 < p < +∞ is
AA − p fp (x) = ( log x) .
5.2.6 Dombi t-norms
The family of Dombi t-norms, introduced by József Dombi (1982), is given for 0 ≤ p ≤ +∞ by
0 ifx = 0 or y = 0 TD(x, y) ifp = 0 T D(x, y) = p T (x, y) ifp = +∞ min 1 1−x p 1−y p 1/p otherwise. 1+(( x ) +( y ) )
D The Dombi t-norm Tp is strict if and only if 0 < p < +∞ (for p = 1 it is the Hamacher product). The family is strictly D increasing and continuous with respect to p. The Dombi t-norm Tp for 0 < p < +∞ arises from the Hamacher product D t-norm by raising its additive generator to the power of p. An additive generator of Tp for 0 < p < +∞ is
( ) 1 − x p f D(x) = . p x
5.2.7 Sugeno–Weber t-norms
The family of Sugeno–Weber t-norms was introduced in the early 1980s by Siegfried Weber; the dual t-conorms were defined already in the early 1970s by Michio Sugeno. It is given for −1 ≤ p ≤ +∞ by 5.3. ORDINAL SUMS 13
− TD(x,( y) ) ifp = 1 T SW(x, y) = max 0, x+y−1+pxy if − 1 < p < +∞ p 1+p Tprod(x, y) ifp = +∞
SW The Sugeno–Weber t-norm Tp is nilpotent if and only if −1 < p < +∞ (for p = 0 it is the Łukasiewicz t-norm). The SW family is strictly increasing and continuous with respect to p. An additive generator of Tp for 0 < p < +∞ [sic] is
{ 1 − x ifp = 0 f SW(x) = p − 1 log1+p(1 + px) otherwise.
5.3 Ordinal sums
The ordinal sum constructs a t-norm from a family of t-norms, by shrinking them into disjoint subintervals of the interval [0, 1] and completing the t-norm by using the minimum on the rest of the unit square. It is based on the following theorem:
Let Ti for i in an index set I be a family of t-norms and (ai, bi) a family of pairwise disjoint (non-empty) open subintervals of [0, 1]. Then the function T: [0, 1]2 → [0, 1] defined as { ( ) x−ai y−ai 2 ai + (bi − ai) · Ti − , − ifx, y ∈ [ai, bi] T (x, y) = bi ai bi ai min(x, y) otherwise
is a t-norm.
Ordinal sum of the Łukasiewicz t-norm on the interval [0.05, 0.45] and the product t-norm on the interval [0.55, 0.95]
The resulting t-norm is called the ordinal sum of the summands (Tᵢ, aᵢ, bᵢ) for i in I, denoted by
⊕ T = (Ti, ai, bi), i∈I or (T1, a1, b1) ⊕ · · · ⊕ (Tn, an, bn) if I is finite. Ordinal sums of t-norms enjoy the following properties: 14 CHAPTER 5. CONSTRUCTION OF T-NORMS
• Each t-norm is a trivial ordinal sum of itself on the whole interval [0, 1].
• The empty ordinal sum (for the empty index set) yields the minimum t-norm T ᵢ . Summands with the mini- mum t-norm can arbitrarily be added or omitted without changing the resulting t-norm.
• It can be assumed without loss of generality that the index set is countable, since the real line can only contain at most countably many disjoint subintervals.
• An ordinal sum of t-norm is continuous if and only if each summand is a continuous t-norm. (Analogously for left-continuity.)
• An ordinal sum is Archimedean if and only if it is a trivial sum of one Archimedean t-norm on the whole unit interval.
• An ordinal sum has zero divisors if and only if for some index i, ai = 0 and Ti has zero divisors. (Analogously for nilpotent elements.) ⊕ If T = i∈I (Ti, ai, bi) is a left-continuous t-norm, then its residuum R is given as follows:
1 ( ) ifx ≤ y x−ai y−ai R(x, y) = ai + (bi − ai) · Ri − , − ifai < y < x ≤ bi bi ai bi ai y otherwise. where Rᵢ is the residuum of Tᵢ, for each i in I.
5.3.1 Ordinal sums of continuous t-norms
The ordinal sum of a family of continuous t-norms is a continuous t-norm. By the Mostert–Shields theorem, every continuous t-norm is expressible as the ordinal sum of Archimedean continuous t-norms. Since the latter are either nilpotent (and then isomorphic to the Łukasiewicz t-norm) or strict (then isomorphic to the product t-norm), each continuous t-norm is isomorphic to the ordinal sum of Łukasiewicz and product t-norms. Important examples of ordinal sums of continuous t-norms are the following ones:
• Dubois–Prade t-norms, introduced by Didier Dubois and Henri Prade in the early 1980s, are the ordinal sums of the product t-norm on [0, p] for a parameter p in [0, 1] and the (default) minimum t-norm on the rest of the unit interval. The family of Dubois–Prade t-norms is decreasing and continuous with respect to p..
• Mayor–Torrens t-norms, introduced by Gaspar Mayor and Joan Torrens in the early 1990s, are the ordinal sums of the Łukasiewicz t-norm on [0, p] for a parameter p in [0, 1] and the (default) minimum t-norm on the rest of the unit interval. The family of Mayor–Torrens t-norms is decreasing and continuous with respect to p..
5.4 Rotations
The construction of t-norms by rotation was introduced by Sándor Jenei (2000). It is based on the following theorem:
Let T be a left-continuous t-norm without zero divisors, N: [0, 1] → [0, 1] the function that assigns 1 − x { | ≤ to x and t = 0.5. Let T1 be the linear transformation of T into [t, 1] and RT1 (x, y) = sup z T1(z, x) y}. Then the function ∈ T1(x, y) ifx, y (t, 1] ∈ ∈ N(RT1 (x, N(y))) ifx (t, 1] and y [0, t] Trot = N(R (y, N(x))) ifx ∈ [0, t] and y ∈ (t, 1] T1 0 ifx, y ∈ [0, t] is a left-continuous t-norm, called the rotation of the t-norm T. 5.5. SEE ALSO 15
The nilpotent minimum as a rotation of the minimum t-norm
Geometrically, the construction can be described as first shrinking the t-norm T to the interval [0.5, 1] and then rotating it by the angle 2π/3 in both directions around the line connecting the points (0, 0, 1) and (1, 1, 0). The theorem can be generalized by taking for N any strong negation, that is, an involutive strictly decreasing continuous function on [0, 1], and for t taking the unique fixed point of N. The resulting t-norm enjoys the following rotation invariance property with respect to N:
T(x, y) ≤ z if and only if T(y, N(z)) ≤ N(x) for all x, y, z in [0, 1].
The negation induced by Tᵣₒ is the function N, that is, N(x) = Rᵣₒ (x, 0) for all x, where Rᵣₒ is the residuum of Tᵣₒ .
5.5 See also
• T-norm
• T-norm fuzzy logics 16 CHAPTER 5. CONSTRUCTION OF T-NORMS
Rotations of the Łukasiewicz, product, nilpotent minimum, and drastic t-norm
5.6 References
• Klement, Erich Peter; Mesiar, Radko; and Pap, Endre (2000), Triangular Norms. Dordrecht: Kluwer. ISBN 0-7923-6416-3.
• Fodor, János (2004), “Left-continuous t-norms in fuzzy logic: An overview”. Acta Polytechnica Hungarica 1(2), ISSN 1785-8860
• Dombi, József (1982), “A general class of fuzzy operators, the DeMorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators”. Fuzzy Sets and Systems 8, 149–163.
• Jenei, Sándor (2000), “Structure of left-continuous t-norms with strong induced negations. (I) Rotation con- struction”. Journal of Applied Non-Classical Logics 10, 83–92.
• Mirko Navara (2007), “Triangular norms and conorms”, Scholarpedia . Chapter 6
Defuzzification
Defuzzification is the process of producing a quantifiable result in fuzzy logic, given fuzzy sets and corresponding membership degrees. It is typically needed in fuzzy control systems. These will have a number of rules that transform a number of variables into a fuzzy result, that is, the result is described in terms of membership in fuzzy sets. For example, rules designed to decide how much pressure to apply might result in “Decrease Pressure (15%), Maintain Pressure (34%), Increase Pressure (72%)". Defuzzification is interpreting the membership degrees of the fuzzy sets into a specific decision or real value. The simplest but least useful defuzzification method is to choose the set with the highest membership, in this case, “Increase Pressure” since it has a 72% membership, and ignore the others, and convert this 72% to some number. The problem with this approach is that it loses information. The rules that called for decreasing or maintaining pressure might as well have not been there in this case. A common and useful defuzzification technique is center of gravity. First, the results of the rules must be added together in some way. The most typical fuzzy set membership function has the graph of a triangle. Now, if this triangle were to be cut in a straight horizontal line somewhere between the top and the bottom, and the top portion were to be removed, the remaining portion forms a trapezoid. The first step of defuzzification typically “chops off” parts of the graphs to form trapezoids (or other shapes if the initial shapes were not triangles). For example, if the output has “Decrease Pressure (15%)", then this triangle will be cut 15% the way up from the bottom. In the most common technique, all of these trapezoids are then superimposed one upon another, forming a single geometric shape. Then, the centroid of this shape, called the fuzzy centroid, is calculated. The x coordinate of the centroid is the defuzzified value.
6.1 Methods
There are many different methods of defuzzification available, including the following:[1]
• AI (adaptive integration)[2] • BADD (basic defuzzification distributions) • BOA (bisector of area) • CDD (constraint decision defuzzification) • COA (center of area) • COG (center of gravity) • ECOA (extended center of area) • EQM (extended quality method) • FCD (fuzzy clustering defuzzification) • FM (fuzzy mean)
17 18 CHAPTER 6. DEFUZZIFICATION
• FOM (first of maximum)
• GLSD (generalized level set defuzzification) • ICOG (indexed center of gravity)
• IV (influence value)[3] • LOM (last of maximum)
• MeOM (mean of maxima) • MOM (middle of maximum)
• QM (quality method) • RCOM (random choice of maximum)
• SLIDE (semi-linear defuzzification)
• WFM (weighted fuzzy mean)
The maxima methods are good candidates for fuzzy reasoning systems. The distribution methods and the area meth- ods exhibit the property of continuity that makes them suitable for fuzzy controllers.[1]
6.2 Notes
[1] van Leekwijck, W.; Kerre, E. E. (1999). “Defuzzification: criteria and classification”. Fuzzy Sets and Systems 108 (2): 159–178. doi:10.1016/S0165-0114(97)00337-0.
[2] Eisele, M.; Hentschel, K. ; Kunemund, T. (1994). “Hardware realization of fast defuzzification by adaptive integration”. Proceedings of the Fourth International Conference on Microelectronics for Neural Networks and Fuzzy Systems 1994: 318– 323. doi:10.1109/ICMNN.1994.593726.
[3] Madau, D. P.; Feldkamp, L. A. (1996). “Influence value defuzzification method”. Fuzzy Systems 3: 1819–1824. doi:10.1109/FUZZY.1996.552647.
6.3 See also
• Fuzzy logic
• Fuzzy set
• Fuzzy control Chapter 7
Degree of truth
In standard mathematics, propositions can typically be considered unambiguously true or false. For instance, the proposition zero belongs to the set { 1 } is regarded as simply false; while the proposition one belongs to the set { 1 } is regarded as simply true. However, some mathematicians, computer scientists, and philosophers have been attracted to the idea that a proposition might be more or less true, rather than simply true or simply false. Consider My coffee is hot. In mathematics, this idea can be developed in terms of fuzzy logic. In computer science, it has found application in artificial intelligence. In philosophy, the idea has proved particularly appealing in the case of vagueness. Degrees of truth is an important concept in law.
7.1 See also
• Artificial intelligence • Bivalence
• Fuzzy logic
• Fuzzy set • Half-truth
• Multi-valued logic • Paradox of the heap
• Truth • Truth value
• Vagueness
7.2 Bibliography
• Zadeh, L.A. (1965). “Fuzzy sets”. Information and Control 8 (3): 338–353. doi:10.1016/S0019-9958(65)90241- X. ISSN 0019-9958.
19 Chapter 8
European Society for Fuzzy Logic and Technology
The European Society for Fuzzy Logic and Technology (EUSFLAT) is a scientific association with the aims to disseminate and promote fuzzy logic and related subjects (sometimes comprised under the collective terms soft computing or computational intelligence) and to provide a platform for exchange between scientists and engineers working in these fields. The society is both open for academic and industrial members.
8.1 History
EUSFLAT was founded in 1998 in Spain as the successor of the National Spanish Fuzzy Logic Society, ESTYLF, with the aim to open the society for members from other European countries. Since then, the society managed to attract a large share of members from outside Spain, and even beyond Europe, with the Spanish members still being the largest group inside EUSFLAT. For these historical reasons, the society is officially registered in Spain.
8.2 Conferences
Starting with 1999, EUSFLAT has been organizing its biannual conferences in odd years. Previous meetings:
• Palma de Mallorca, Balearic Islands, Spain, September 22–25, 1999 (jointly with National Spanish conference, ESTYLF)
• Leicester, United Kingdom, September 5–7, 2001
• Zittau, Germany, September 10–12, 2003
• Barcelona, Catalonia, Spain, September 7–9, 2005 (jointly with 11th Rencontres Francophones sur la Logique Floue et ses Applications)
• Ostrava, Czech Republic, September 11–14, 2007
• Lisbon, Portugal, July 20–24, 2009 (jointly with 13th World Congress of the International Fuzzy Systems Association)
• Aix-les-Bains, France, July 18–22, 2011 (jointly with Les Rencontres Francophones sur la Logique Floue et ses Applications)
• Milan, Italy, September 11–13, 2013
• Gijón, Spain, June, 30–3 July 2015
20 8.3. PUBLICATIONS 21
8.3 Publications
• EUSFLAT publishes the proceedings of its conferences in an open access manner.[1]
• Until 2010, Mathware & Soft Computing was the official journal of EUSFLAT. On July 1, 2010, the International Journal of Computational Intelligence Systems (Atlantis Press, ISSN 75-6891 (print) / ISSN 1875-6883 (on- line)) became the official journal of EUSFLAT.
• EUSFLAT publishes an electronic newsletter with three issues a year.
8.4 Presidents
EUSFLAT is led by the President, who is elected for a two-year period, and cannot serve for more than two consecutive periods.[2]
• Francesc Esteva (1998–2011)
• Luis Magdalena (2001–2005) • Ulrich Bodenhofer (2005–2009)
• Javier Montero (2009–2013) • Gabriella Pasi (2013–present)
8.5 References
[1] EUSFLAT Conference Proceedings
[2] EUSFLAT bylaws
8.6 External links
• The EUSFLAT website Chapter 9
Fuzzy architectural spatial analysis
Fuzzy architectural spatial analysis (FASA) (also fuzzy inference system (FIS) based architectural space analysis or fuzzy spatial analysis) is a spatial analysis method of analyzing the spatial formation and architectural space intensity within any architectural organization.[1] Fuzzy architectural spatial analysis is used in architecture, interior design, urban planning and similar spatial design fields.
9.1 Overview
Fuzzy architectural spatial analysis was developed from the architectural theory of space syntax[2][3] and visibility graph analysis,[4] by Burcin Cem Arabacioglu (2010), and is applied with the help of a fuzzy system with a Mamdami inference system based on fuzzy logic within any architectural space. Fuzzy architectural spatial analysis model analyses the space by considering the perceivable architectural element by their boundary and stress characteristics and intensity properties. The method is capable of taking all sensorial factors into account during analyses in conformably with the perception process of architectural space which is a multi-sensored act.
9.2 References
[1] Arabacioglu, Burcin Cem (2010). “Using fuzzy inference system for architectural space analysis”. Applied Soft Computing 10 (3): 926–937. doi:10.1016/j.asoc.2009.10.011
[2] Hillier, Bill and Hanson, Julienne (1984), “The Social Logic of Space”, Cambridge University Press: Cambridge.
[3] Hillier, Bill (1999), “Space is the Machine: A Configurational Theory of Architecture”, Cambridge University Press: Cambridge.
[4] Turner, Alasdair; Doxa, Maria; O'Sullivan, David and Penn, Alan (2001). “From isovists to visibility graphs: a methodology for the analysis of architectural space”. Environment and Planning B 28 (1): 103–121. doi:10.1068/b2684
9.3 Further reading
• Arabacioglu, Burcin Cem (2010). “Using fuzzy inference system for architectural space analysis”. Applied Soft Computing 10 (3): 926–937. doi:10.1016/j.asoc.2009.10.011.
• Cekmis, Asli; Hacihasanoglu, Isis; Ostwald, Michael J (2013), “A computational model for accommodat- ing spatial uncertainty: Predicting inhabitation patterns in open-planned spaces”, Building and Environment, doi:10.1016/j.buildenv.2013.11.023.
• Dutta, Kamlesh; Sarthak, Siddhant (2011), “Architectural space planning using evolutionary computing ap- proaches: a review”, Artificial Intelligence Review 36 (4): 311–321, doi:10.1007/s10462-011-9217-y.
22 9.4. SEE ALSO 23
• Indraprastha, Aswin; Shinozaki, Michihiko (2011), “Elaboration Model for Mapping Architectural Space”, Journal of Asian Architecture and Building Engineering 10 (2): 1–8 • Lin, Yuan Horng (2013), “Fuzzy Kappa Coefficient with Simulated Comparisons”, Applied Mechanics and Materials, 303-306: 372–375, doi:10.4028/www.scientific.net/AMM.303-306.372 • Wurzer, Gabriel (2013), “In-process agent simulation for early stages of hospital planning”, Mathematical and Computer Modelling of Dynamical Systems: Methods, Tools and Applications in Engineering and Related Sciences 19 (4): 331–343, doi:10.1080/13873954.2012.761638.
• Yang, Xin; Xu, Duan-qing; Zhao, Lei (2013), “Efficient data management for incoherent ray tracing”, Applied Soft Computing 13 (1): 1–8, doi:10.1016/j.asoc.2012.07.002.
9.4 See also
• Spatial analysis
• Space syntax • Spatial network analysis software
• Visibility graph • Visibility graph analysis
• Boundary problem (in spatial analysis) Chapter 10
Fuzzy associative matrix
A fuzzy associative matrix expresses fuzzy logic rules in tabular form. These rules usually take two variables as input, mapping cleanly to a two-dimensional matrix, although theoretically a matrix of any number of dimensions is possible. Suppose a professional is tasked with writing fuzzy logic rules for a video game monster. In the game being built, entities have two variables: hit points (HP) and firepower (FP): This translates to: IF MonsterHP IS VeryLowHP AND MonsterFP IS VeryWeakFP THEN Retreat IF MonsterHP IS LowHP AND MonsterFP IS VeryWeakFP THEN Retreat IF MonsterHP IS MediumHP AND MonsterFP is VeryWeakFP THEN Defend Multiple rules can fire at once, and often will, because the distinction between “very low” and “low” is fuzzy. If it is more “very low” than it is low, then the “very low” rule will generate a stronger response. The program will evaluate all the rules that fire and use an appropriate defuzzification method to generate its actual response. An implementation of this system might use either the matrix or the explicit IF/THEN form. The matrix makes it easy to visualize the system, but it also makes it impossible to add a third variable just for one rule, so it is less flexible. There is no inherent pattern in the matrix. It appears as if the rules were just made up, and indeed they were. This is both a strength and a weakness of fuzzy logic in general. It is often impractical or impossible to find an exact set of rules or formulae for dealing with a specific situation. For a sufficiently complex game, a mathematician would not be able to study the system and figure out a mathematically accurate set of rules. However, this weakness is intrinsic to the realities of the situation, not of fuzzy logic itself. The strength of the system is that even if one of the rules is wrong, even greatly wrong, other rules that are correct are likely to fire as well and they may compensate for the error. This does not mean a fuzzy system should be sloppy. Depending on the system, it might get away with being sloppy, but it will underperform. While the rules are fairly arbitrary, they should be chosen carefully. If possible, an expert should decide on the rules, and the sets and rules should be tested vigorously and refined as needed. In this way, a fuzzy system is like an expert system. (Fuzzy logic is used in many true expert systems, as well.)
24 Chapter 11
Fuzzy classification
Fuzzy classification is the process of grouping elements into a fuzzy set[1] whose membership function is defined by the truth value of a fuzzy propositional function.[2][3][4] A fuzzy class ~C = { i | ~Π(i) } is defined as a fuzzy set ~C of individuals i satisfying a fuzzy classification predicate ~Π which is a fuzzy propositional function. The domain of the fuzzy class operator ~{ .| .} is the set of variables V and the set of fuzzy propositional functions ~PF, and the range is the fuzzy powerset (the set of fuzzy subsets) of this universe, ~P(U): ~{ .| .}∶V × ~PF ⟶ ~P(U) A fuzzy propositional function is, analogous to,[5] an expression containing one or more variables, such that, when values are assigned to these variables, the expression becomes a fuzzy proposition in the sense of.[6] Accordingly, fuzzy classification is the process of grouping individuals having the same characteristics into a fuzzy set. A fuzzy classification corresponds to a membership function μ that indicates whether an individual is a member of a class, given its fuzzy classification predicate ~Π. μ∶~PF × U ⟶ ~T Here, ~T is the set of fuzzy truth values (the interval between zero and one). The fuzzy classification predicate ~Π corresponds to a fuzzy restriction “i is R” [6] of U, where R is a fuzzy set defined by a truth function. The degree of membership of an individual i in the fuzzy class ~C is defined by the truth value of the corresponding fuzzy predicate. μ~C(i):= τ(~Π(i))
11.1 Classification
Intuitively, a class is a set that is defined by a certain property, and all objects having that property are elements of that class. The process of classification evaluates for a given set of objects whether they fulfill the classification property, and consequentially are a member of the corresponding class. However, this intuitive concept has some logical subtleties that need clarification. A class logic[7] is a logical system which supports set construction using logical predicates with the class operator { .| .}. A class C = { i | Π(i) } is defined as a set C of individuals i satisfying a classification predicate Π which is a propositional function. The domain of the class operator { .| .} is the set of variables V and the set of propositional functions PF, and the range is the powerset of this universe P(U) that is, the set of possible subsets: { .| .} ∶V×PF⟶P(U) Here is an explanation of the logical elements that constitute this definition:
• An individual is a real object of reference. • A universe of discourse is the set of all possible individuals considered.
25 26 CHAPTER 11. FUZZY CLASSIFICATION
• A variable V:⟶R is a function which maps into a predefined range R without any given function arguments: a zero-place function. • A propositional function is “an expression containing one or more undetermined constituents, such that, when values are assigned to these constituents, the expression becomes a proposition”.[5]
In contrast, classification is the process of grouping individuals having the same characteristics into a set. A classifi- cation corresponds to a membership function μ that indicates whether an individual is a member of a class, given its classification predicate Π. μ∶PF × U ⟶ T The membership function maps from the set of propositional functions PF and the universe of discourse U into the set of truth values T. The membership μ of individual i in Class C is defined by the truth value τ of the classification predicate Π. μC(i):=τ(Π(i)) In classical logic the truth values are certain. Therefore a classification is crisp, since the truth values are either exactly true or exactly false.
11.2 See also
• Fuzzy logic
11.3 References
[1] Zadeh, L. A. (1965). Fuzzy sets. Information and Control (8), pp. 338–353.
[2] Zimmermann, H.-J. (2000). Practical Applications of Fuzzy Technologies. Springer.
[3] Meier, A., Schindler, G., & Werro, N. (2008). Fuzzy classification on relational databases. In M. Galindo (Hrsg.), Hand- book of research on fuzzy information processing in databases (Bd. II, S. 586-614). Information Science Reference.
[4] Del Amo, A., Montero, J., & Cutello, V. (1999). On the principles of fuzzy classification. Proc. 18th North American Fuzzy Information Processing Society Annual Conf, (S. 675 – 679).
[5] Russel, B. (1919). Introduction to Mathematical Philosophy. London: George Allen & Unwin, Ltd., S. 155
[6] Zadeh, L. A. (1975). Calculus of fuzzy restrictions. In L. A. Zadeh, K.-S. Fu, K. Tanaka, & M. Shimura (Hrsg.), Fuzzy sets and Their Applications to Cognitive and Decision Processes. New York: Academic Press.
[7] Glubrecht, J.-M., Oberschelp, A., & Todt, G. (1983). Klassenlogik. Mannheim/Wien/Zürich: Wissenschaftsverlag. Chapter 12
Fuzzy cognitive map
A Fuzzy cognitive map is a cognitive map within which the relations between the elements (e.g. concepts, events, project resources) of a “mental landscape” can be used to compute the “strength of impact” of these elements. Ron Axelord introduced Cognitive Maps as a formal way of representing social scientific knowledge and modeling decision making in social and political systems. Then brought in the computation fuzzy logic.[1]
12.1 Details
Fuzzy cognitive maps are signed fuzzy digraphs. They may look at first blush like Hasse diagrams but they are not. Spreadsheets or tables are used to map FCMs into matrices for further computation.[2][3][4] FCM is a technique used for causal knowledge acquisition and representation, it supports causal knowledge reasoning process and belong to the neuro-fuzzy system that aim at solving decision making problems, modeling and simulate complex systems. Learning algorithms have been proposed for training and updating FCMs weights mostly based on ideas coming from the field of Artificial Neural Networks. Adaptation and learning methodologies used to adapt the FCM model and adjust its weights. Kosko and Dickerson (Dickerson & Kosko, 1994) suggested the Differential Hebbian Learning (DHL) to train FCM. There have been proposed algorithms based on the initial Hebbian algorithm; others algorithms come from the field of genetic algorithms, swarm intelligence and evolutionary computation. Learning algorithms are used to overcome the shortcomings that the traditional FCM present i.e. decreasing the human intervention by suggested automated FCM candidates; or by activating only the most relevant concepts every execution time; or by making models more transparent and dynamic. . Fuzzy cognitive maps (FCMs) have gained considerable research interest due to their ability in representing structured knowledge and model complex systems in various fields. This growing interest led to the need for enhancement and making more reliable models that can better represent real situations. A first simple application of FCMs is described in a book[5] of William R. Taylor, where the war in Afghanistan and Iraq is analyzed. And in Bart Kosko's book Fuzzy Thinking,[6] several Hasse diagrams illustrate the use of FCMs. As an example, one FCM quoted from Rod Taber[7] describes 11 factors of the American cocaine market and the relations between these factors. For computations, Taylor uses pentavalent logic (scalar values out of {−1,−0.5,0,+0.5,+1}). That particular map of Taber uses trivalent logic (scalar values out of {−1,0,+1}). Taber et al. also illustrate the dynamics of map fusion and give a theorem on the convergence of combination in a related article [8] While applications in social sciences[5][6][7][9] introduced FCMs to the public, they are used in a much wider range of applications, which all have to deal with creating and using models[10] of uncertainty and complex processes and systems. Examples:
• In business FCMs can be used for product planning.[11]
• In economics, FCMs support the use of game theory in more complex settings.[12]
• In Medical applications to model systems, provide diagnosis , develop decision support systems and medical assessment.
• In Engineering for modeling and control mainly of complex systems
27 28 CHAPTER 12. FUZZY COGNITIVE MAP
Rod Tabers FCM depicting eleven factors of the American drug market
• In project planning FCMs help to analyze the mutual dependencies between project resources.
• In robotics[6][13] FCMs support machines to develop fuzzy models of their environments and to use these models to make crisp decisions.
• In computer assisted learning FCMs enable computers to check whether students understand their lessons.[14]
• In expert systems[7] a few or many FCMs can be aggregated into one FCM in order to process estimates of knowledgeable persons.[15]
• In IT project management, a FCM-based methodology helps to success modelling.[16]
FCMappers[17] - an international online community for the analysis and the visualization of fuzzy cognitive maps offer support for starting with FCM and also provide a MS-Excel based tool that is able to check and analyse FCMs. 12.2. REFERENCES 29
The output is saved as Pajek file and can be visualized within 3rd party software like Pajek, Visone,... . They also offer to adapt the software to specific research needs. On their webpage you also will find a linklist for interesting sci- entific articles, related software, institutes, people and projects. The FCMappers have about one thousand registered members worldwide. Additional FCM software tools, such as Mental Modeler,[18][19] have recently been developed as a decision-support tool for use in social science research, collaborative decision-making, and natural resource planning.
12.2 References
[1] Bart Kosko, Fuzzy Cognitive Maps, International Journal of Man-Machine Studies, 24(1986) 65-75 (first introduction of FCMs): see also
[2] FCMapper - Excel based FCM analysis and visualization tool: http://www.FCMappers.net/joomla/index.php?option= com_content&view=article&id=52&Itemid=53
[3] On line calculator and downloadable Java applications for FCM computations: http://www.ochoadeaspuru.com/fuzcogmap/ index.php
[4] Java standalone library for FCM computations: http://jfcm.megadix.it/
[5] William R. Taylor: Lethal American Confusion (How Bush and the Pacifists Each Failed in the War on Terrorism), 2006, ISBN 0-595-40655-6 (FCM application in chapter 14)
[6] Bart Kosko: Fuzzy Thinking, 1993/1995, ISBN 0-7868-8021-X (Chapter 12: Adaptive Fuzzy Systems)
[7] Rod Taber: Knowledge Processing with Fuzzy Cognitive Maps, Expert Systems with Applications, vol. 2, no. 1, 83-87, 1991 (Hasse diagram in German Wikipedia)
[8] Rod Taber, Ronald R. Yager, and Cathy M. Helgason:Quantization Effects on the Equilibrium Behavior of Combined Fuzzy Cognitive Maps, International Journal of Intelligent Systems, vol. 22, 181-202, 2007.
[9] Costas Neocleous, Christos Schizas, Costas Yenethlis: Fuzzy Cognitive Models in Studying Political Dynamics - The case of the Cyprus problem
[10] Chrysostomos D. Stylios, Voula C. Georgopoulos, Peter P. Groumpos: The Use of Fuzzy Cognitive Maps in Modeling Systems
[11] Antonie Jetter: Produktplanung im Fuzzy Front End, 2005, ISBN 3-8350-0144-2
[12] Vesa A. Niskanen: Application of Fuzzy Linguistic Cognitive Maps to Prisoner’s Dilemma, 2005, ICIC International pp. 139-152, ISSN 1349-4198
[13] Marc Böhlen: More Robots in Cages,
[14] Benjoe A. Juliano, Wylis Bandler: Tracing Chains-of-Thought (Fuzzy Methods in Cognitive Diagnosis), Physica-Verlag Heidelberg 1996, ISBN 3-7908-0922-5
[15] W. B. Vasantha Kandasamy, Florentin Smarandache: Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps, 2003, ISBN 1-931233-76-4
[16] L. Rodriguez-Repiso, R. Setchi, and J.L. Salmeron. Modelling IT Projects success with Fuzzy Cognitive Maps. Expert Systems with Applications 32(2) pp. 543-559. 2007.
[17] FCMappers - international community for fuzzy cognitive mapping: http://www.FCMappers.net/
[18] Gray, S. Gray, S., Cox, L., and Henly-Shepard, S. 2013 Mental modeler: A fuzzy-logic cognitive mapping modeling tool for adaptive environmental management. Proceedings of the 46th International Conference on Complex Systems. 963-973. http://www.computer.org/csdl/proceedings/hicss/2013/4892/00/4892a965.pdf
[19] http://www.mentalmodeler.com/
12.3 External links
Soft Computing Chapter 13
Fuzzy Control Language
Fuzzy Control Language, or FCL, is a language for implementing fuzzy logic, especially fuzzy control. It was standardized by IEC 61131-7. It is a domain-specific programming language: it has no features unrelated to fuzzy logic, so it is impossible to even print "Hello, world!". Therefore, one does not write a program in FCL, but one may write part of it in FCL. RULE 0: IF (Temperature IS Cold) THEN (Output IS High) FCL is not an entirely complete fuzzy language, for instance, it does not support “hedges”, which are adverbs that modify the set. For instance, the programmer cannot write: RULE 0: IF (Temperature IS VERY Cold) THEN (Output IS VERY High) However, the programmer can simply define new sets for “very cold” and “very high”. FCL also lacks support for higher-order fuzzy sets, subsets, and so on. None of these features are essential to fuzzy control, although they may be nice to have.
13.1 External links
• fuzzyTECH, a commercial fuzzy logic development system containing the specification document for IEC1131- 7 (select Fuzzy Application Library) • IEC 1131-7 CD1 IEC 1131-7 CD1 PDF
• fuzzylite, A fuzzy logic controller library written in C++. • Free Fuzzy Logic Library (FFLL), an implementation library written in C++.
• JFuzzyLogic, open source FCL + Fuzzy Logic Package (sourceforge, java) • AwiFuzz, open source implementation written in C++ covering all three levels of IEC 61131-7
Fuzzy Controller Language IEC 1131-7 CD1
30 Chapter 14
Fuzzy control system
“Fuzzy control” and “Fuzzy Control” redirect here. For the rock band, see Fuzzy Control (band).
A fuzzy control system is a control system based on fuzzy logic—a mathematical system that analyzes analog input values in terms of logical variables that take on continuous values between 0 and 1, in contrast to classical or digital logic, which operates on discrete values of either 1 or 0 (true or false, respectively).
14.1 Overview
Fuzzy logic is widely used in a machine control. The term “fuzzy” refers to the fact that the logic involved can deal with concepts that cannot be expressed as the “true” or “false” but rather as “partially true”. Although alternative approaches such as genetic algorithms and neural networks can perform just as well as fuzzy logic in many cases, fuzzy logic has the advantage that the solution to the problem can be cast in terms that human operators can understand, so that their experience can be used in the design of the controller. This makes it easier to mechanize tasks that are already successfully performed by humans.
14.2 History and applications
Fuzzy logic was first proposed by Lotfi A. Zadeh of the University of California at Berkeley in a 1965 paper. He elaborated on his ideas in a 1973 paper that introduced the concept of “linguistic variables”, which in this article equates to a variable defined as a fuzzy set. Other research followed, with the first industrial application, a cement kiln built in Denmark, coming on line in 1975. Fuzzy systems were initially implemented in Japan.
• Interest in fuzzy systems was sparked by Seiji Yasunobu and Soji Miyamoto of Hitachi, who in 1985 provided simulations that demonstrated the feasibility of fuzzy control systems for the Sendai railway. Their ideas were adopted, and fuzzy systems were used to control accelerating, braking, and stopping when the line opened in 1987.
• In 1987, Takeshi Yamakawa demonstrated the use of fuzzy control, through a set of simple dedicated fuzzy logic chips, in an "inverted pendulum" experiment. This is a classic control problem, in which a vehicle tries to keep a pole mounted on its top by a hinge upright by moving back and forth. Yamakawa subsequently made the demonstration more sophisticated by mounting a wine glass containing water and even a live mouse to the top of the pendulum: the system maintained stability in both cases. Yamakawa eventually went on to organize his own fuzzy-systems research lab to help exploit his patents in the field.
• Japanese engineers subsequently developed a wide range of fuzzy systems for both industrial and consumer applications. In 1988 Japan established the Laboratory for International Fuzzy Engineering (LIFE), a coop- erative arrangement between 48 companies to pursue fuzzy research. The automotive company Volkswagen was the only foreign corporate member of LIFE, dispatching a researcher for a duration of three years.
31 32 CHAPTER 14. FUZZY CONTROL SYSTEM
• Japanese consumer goods often incorporate fuzzy systems. Matsushita vacuum cleaners use microcontrollers running fuzzy algorithms to interrogate dust sensors and adjust suction power accordingly. Hitachi washing machines use fuzzy controllers to load-weight, fabric-mix, and dirt sensors and automatically set the wash cycle for the best use of power, water, and detergent.
• Canon developed an autofocusing camera that uses a charge-coupled device (CCD) to measure the clarity of the image in six regions of its field of view and use the information provided to determine if the image is in focus. It also tracks the rate of change of lens movement during focusing, and controls its speed to prevent overshoot. The camera’s fuzzy control system uses 12 inputs: 6 to obtain the current clarity data provided by the CCD and 6 to measure the rate of change of lens movement. The output is the position of the lens. The fuzzy control system uses 13 rules and requires 1.1 kilobytes of memory.
• An industrial air conditioner designed by Mitsubishi uses 25 heating rules and 25 cooling rules. A temperature sensor provides input, with control outputs fed to an inverter, a compressor valve, and a fan motor. Compared to the previous design, the fuzzy controller heats and cools five times faster, reduces power consumption by 24%, increases temperature stability by a factor of two, and uses fewer sensors.
• Other applications investigated or implemented include: character and handwriting recognition; optical fuzzy systems; robots, including one for making Japanese flower arrangements; voice-controlled robot helicopters (hovering is a “balancing act” rather similar to the inverted pendulum problem); control of flow of powders in film manufacture; elevator systems; and so on.
Work on fuzzy systems is also proceeding in the United State and Europe, although on a less extensive scale than in Japan.
• The US Environmental Protection Agency has investigated fuzzy control for energy-efficient motors, and NASA has studied fuzzy control for automated space docking: simulations show that a fuzzy control system can greatly reduce fuel consumption.
• Firms such as Boeing, General Motors, Allen-Bradley, Chrysler, Eaton, and Whirlpool have worked on fuzzy logic for use in low-power refrigerators, improved automotive transmissions, and energy-efficient electric mo- tors.
• In 1995 Maytag introduced an “intelligent” dishwasher based on a fuzzy controller and a “one-stop sensing module” that combines a thermistor, for temperature measurement; a conductivity sensor, to measure detergent level from the ions present in the wash; a turbidity sensor that measures scattered and transmitted light to measure the soiling of the wash; and a magnetostrictive sensor to read spin rate. The system determines the optimum wash cycle for any load to obtain the best results with the least amount of energy, detergent, and water. It even adjusts for dried-on foods by tracking the last time the door was opened, and estimates the number of dishes by the number of times the door was opened.
Research and development is also continuing on fuzzy applications in software, as opposed to firmware, design, including fuzzy expert systems and integration of fuzzy logic with neural-network and so-called adaptive "genetic" software systems, with the ultimate goal of building “self-learning” fuzzy-control systems.
14.3 Fuzzy sets
See also: fuzzy set
The input variables in a fuzzy control system are in general mapped by sets of membership functions similar to this, known as “fuzzy sets”. The process of converting a crisp input value to a fuzzy value is called “fuzzification”. A control system may also have various types of switch, or “ON-OFF”, inputs along with its analog inputs, and such switch inputs of course will always have a truth value equal to either 1 or 0, but the scheme can deal with them as simplified fuzzy functions that happen to be either one value or another. 14.3. FUZZY SETS 33
Given "mappings" of input variables into membership functions and truth values, the microcontroller then makes decisions for what action to take, based on a set of “rules”, each of the form: IF brake temperature IS warm AND speed IS not very fast THEN brake pressure IS slightly decreased. In this example, the two input variables are “brake temperature” and “speed” that have values defined as fuzzy sets. The output variable, “brake pressure” is also defined by a fuzzy set that can have values like “static” or “slightly increased” or “slightly decreased” etc. This rule by itself is very puzzling since it looks like it could be used without bothering with fuzzy logic, but remember that the decision is based on a set of rules:
• All the rules that apply are invoked, using the membership functions and truth values obtained from the inputs, to determine the result of the rule.
• This result in turn will be mapped into a membership function and truth value controlling the output variable.
• These results are combined to give a specific (“crisp”) answer, the actual brake pressure, a procedure known as "defuzzification".
This combination of fuzzy operations and rule-based "inference" describes a “fuzzy expert system”. Traditional control systems are based on mathematical models in which the control system is described using one or more differential equations that define the system response to its inputs. Such systems are often implemented as “PID controllers” (proportional-integral-derivative controllers). They are the products of decades of development and theoretical analysis, and are highly effective. If PID and other traditional control systems are so well-developed, why bother with fuzzy control? It has some advantages. In many cases, the mathematical model of the control process may not exist, or may be too “expensive” in terms of computer processing power and memory, and a system based on empirical rules may be more effective. Furthermore, fuzzy logic is well suited to low-cost implementations based on cheap sensors, low-resolution analog-to- digital converters, and 4-bit or 8-bit one-chip microcontroller chips. Such systems can be easily upgraded by adding new rules to improve performance or add new features. In many cases, fuzzy control can be used to improve existing traditional controller systems by adding an extra layer of intelligence to the current control method.
14.3.1 Fuzzy control in detail
Fuzzy controllers are very simple conceptually. They consist of an input stage, a processing stage, and an output stage. The input stage maps sensor or other inputs, such as switches, thumbwheels, and so on, to the appropriate membership functions and truth values. The processing stage invokes each appropriate rule and generates a result for each, then combines the results of the rules. Finally, the output stage converts the combined result back into a specific control output value. The most common shape of membership functions is triangular, although trapezoidal and bell curves are also used, but the shape is generally less important than the number of curves and their placement. From three to seven curves are generally appropriate to cover the required range of an input value, or the "universe of discourse" in fuzzy jargon. As discussed earlier, the processing stage is based on a collection of logic rules in the form of IF-THEN statements, where the IF part is called the “antecedent” and the THEN part is called the “consequent”. Typical fuzzy control systems have dozens of rules. Consider a rule for a thermostat: IF (temperature is “cold”) THEN (heater is “high”) This rule uses the truth value of the “temperature” input, which is some truth value of “cold”, to generate a result in the fuzzy set for the “heater” output, which is some value of “high”. This result is used with the results of other rules to finally generate the crisp composite output. Obviously, the greater the truth value of “cold”, the higher the truth value of “high”, though this does not necessarily mean that the output itself will be set to “high” since this is only one rule among many. In some cases, the membership functions can be modified by “hedges” that are equivalent to adverbs. Common hedges include “about”, “near”, “close to”, “approximately”, “very”, “slightly”, “too”, “extremely”, and “somewhat”. These operations may have precise definitions, though the definitions can vary considerably between different implementations. “Very”, for one example, squares membership functions; since the membership values are 34 CHAPTER 14. FUZZY CONTROL SYSTEM always less than 1, this narrows the membership function. “Extremely” cubes the values to give greater narrowing, while “somewhat” broadens the function by taking the square root. In practice, the fuzzy rule sets usually have several antecedents that are combined using fuzzy operators, such as AND, OR, and NOT, though again the definitions tend to vary: AND, in one popular definition, simply uses the minimum weight of all the antecedents, while OR uses the maximum value. There is also a NOT operator that subtracts a membership function from 1 to give the “complementary” function. There are several ways to define the result of a rule, but one of the most common and simplest is the “max-min” inference method, in which the output membership function is given the truth value generated by the premise. Rules can be solved in parallel in hardware, or sequentially in software. The results of all the rules that have fired are “defuzzified” to a crisp value by one of several methods. There are dozens, in theory, each with various advantages or drawbacks. The “centroid” method is very popular, in which the “center of mass” of the result provides the crisp value. Another approach is the “height” method, which takes the value of the biggest contributor. The centroid method favors the rule with the output of greatest area, while the height method obviously favors the rule with the greatest output value. The diagram below demonstrates max-min inferencing and centroid defuzzification for a system with input variables “x”, “y”, and “z” and an output variable “n”. Note that “mu” is standard fuzzy-logic nomenclature for “truth value":
Notice how each rule provides a result as a truth value of a particular membership function for the output variable. In centroid defuzzification the values are OR'd, that is, the maximum value is used and values are not added, and the results are then combined using a centroid calculation. Fuzzy control system design is based on empirical methods, basically a methodical approach to trial-and-error. The general process is as follows: 14.3. FUZZY SETS 35
• Document the system’s operational specifications and inputs and outputs. • Document the fuzzy sets for the inputs. • Document the rule set. • Determine the defuzzification method. • Run through test suite to validate system, adjust details as required. • Complete document and release to production.
As a general example, consider the design of a fuzzy controller for a steam turbine. The block diagram of this control system appears as follows: The input and output variables map into the following fuzzy set:
—where: N3: Large negative. N2: Medium negative. N1: Small negative. Z: Zero. P1: Small positive. P2: Medium positive. P3: Large positive. The rule set includes such rules as: rule 1: IF temperature IS cool AND pressure IS weak, THEN throttle is P3. rule 2: IF temperature IS cool AND pressure IS low, THEN throttle is P2. rule 3: IF temperature IS cool AND pressure IS ok, THEN throttle is Z. rule 4: IF temperature IS cool AND pressure IS strong, THEN throttle is N2. 36 CHAPTER 14. FUZZY CONTROL SYSTEM
In practice, the controller accepts the inputs and maps them into their membership functions and truth values. These mappings are then fed into the rules. If the rule specifies an AND relationship between the mappings of the two input variables, as the examples above do, the minimum of the two is used as the combined truth value; if an OR is specified, the maximum is used. The appropriate output state is selected and assigned a membership value at the truth level of the premise. The truth values are then defuzzified. For an example, assume the temperature is in the “cool” state, and the pressure is in the “low” and “ok” states. The pressure values ensure that only rules 2 and 3 fire:
The two outputs are then defuzzified through centroid defuzzification: ______| Z P2 1 -+ * * | * * * * | * * * * 14.3. FUZZY SETS 37
| * * * * | * 222222222 | * 22222222222 | 333333332222222222222 +--−33333333222222222222222--> ^ +150 ______The output value will adjust the throttle and then the control cycle will begin again to generate the next value .
14.3.2 Building a fuzzy controller
Consider implementing with a microcontroller chip a simple feedback controller:
A fuzzy set is defined for the input error variable “e”, and the derived change in error, “delta”, as well as the “output”, as follows: LP: large positive SP: small positive ZE: zero SN: small negative LN: large negative If the error ranges from −1 to +1, with the analog-to-digital converter used having a resolution of 0.25, then the input variable’s fuzzy set (which, in this case, also applies to the output variable) can be described very simply as a table, with the error / delta / output values in the top row and the truth values for each membership function arranged in rows beneath: ______−1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 1 ______mu(LP) 0 0 0 0 0 0 0.3 0.7 1 mu(SP) 0 0 0 0 0.3 0.7 1 0.7 0.3 mu(ZE) 0 0 0.3 0.7 1 0.7 0.3 0 0 mu(SN) 0.3 0.7 1 0.7 0.3 0 0 0 0 mu(LN) 1 0.7 0.3 0 0 0 0 0 0 ______— or, in graphical form (where each “X” has a value of 0.1): LN SN ZE SP LP +------+ | | −1.0 | XXXXXXXXXX XXX : : : | −0.75 | XXXXXXX XXXXXXX : : : | −0.5 | XXX XXXXXXXXXX XXX : : | −0.25 | : XXXXXXX XXXXXXX : : | 0.0 | : XXX XXXXXXXXXX XXX : | 0.25 | : : XXXXXXX XXXXXXX : | 0.5 | : : XXX XXXXXXXXXX XXX | 0.75 | : : : XXXXXXX XXXXXXX | 1.0 | : : : XXX XXXXXXXXXX | | | +------+ Suppose this fuzzy system has the following rule base: rule 1: IF e = ZE AND delta = ZE THEN output = ZE rule 2: IF e = ZE AND delta = SP THEN output = SN rule 3: IF e = SN AND delta = SN THEN output = LP rule 4: IF e = LP OR delta = LP THEN output = LN These rules are typical for control applications in that the antecedents consist of the logical combination of the error and error-delta signals, while the consequent is a control command output. The rule outputs can be defuzzified using a discrete centroid computation: SUM( I = 1 TO 4 OF ( mu(I) * output(I) ) ) / SUM( I = 1 TO 4 OF mu(I) ) Now, suppose that at a given time we have: e = 0.25 delta = 0.5 Then this gives: ______e delta ______mu(LP) 0 0.3 mu(SP) 0.7 1 mu(ZE) 0.7 0.3 mu(SN) 0 0 mu(LN) 0 0 ______Plugging this into rule 1 gives: rule 1: IF e = ZE AND delta = ZE THEN output = ZE mu(1) = MIN( 0.7, 0.3 ) = 0.3 output(1) = 0 -- where:
• mu(1): Truth value of the result membership function for rule 1. In terms of a centroid calculation, this is the “mass” of this result for this discrete case.
• output(1): Value (for rule 1) where the result membership function (ZE) is maximum over the output variable fuzzy set range. That is, in terms of a centroid calculation, the location of the “center of mass” for this individual 38 CHAPTER 14. FUZZY CONTROL SYSTEM
result. This value is independent of the value of “mu”. It simply identifies the location of ZE along the output range.
The other rules give: rule 2: IF e = ZE AND delta = SP THEN output = SN mu(2) = MIN( 0.7, 1 ) = 0.7 output(2) = −0.5 rule 3: IF e = SN AND delta = SN THEN output = LP mu(3) = MIN( 0.0, 0.0 ) = 0 output(3) = 1 rule 4: IF e = LP OR delta = LP THEN output = LN mu(4) = MAX( 0.0, 0.3 ) = 0.3 output(4) = −1 The centroid computation yields: mu(1).output(1)+mu(2).output(2)+mu(3).output(3)+mu(4).output(4) (0.3∗0)+(0.7∗−0.5)+(0∗1)+(0.3∗−1) − mu(1)+mu(2)+mu(3)+mu(4) = 0.3+0.7+0+0.3 = 0.5 — for the final control output. Simple. Of course the hard part is figuring out what rules actually work correctly in practice. If you have problems figuring out the centroid equation, remember that a centroid is defined by summing all the moments (location times mass) around the center of gravity and equating the sum to zero. So if X0 is the center of gravity, Xi is the location of each mass, and Mi is each mass, this gives:
0 = (X1 − X0) ∗ M1 + (X2 − X0) ∗ M2 + ... + (Xn − X0) ∗ Mn 0 = (X1 ∗ M1 + X2 ∗ M2 + ... + Xn ∗ Mn) − X0 ∗ (M1 + M2 + ... + Mn) X0 ∗ (M1 + M2 + ... + Mn) = X1 ∗ M1 + X2 ∗ M2 + ... + Xn ∗ Mn X1∗M1+X2∗M2+...+Xn∗Mn X0 = M1+M2+...+Mn In our example, the values of mu correspond to the masses, and the values of X to location of the masses (mu, however, only 'corresponds to the masses’ if the initial 'mass’ of the output functions are all the same/equivalent. If they are not the same, i.e. some are narrow triangles, while others maybe wide trapizoids or shouldered triangles, then the mass or area of the output function must be known or calculated. It is this mass that is then scaled by mu and multiplied by its location X_i). This system can be implemented on a standard microprocessor, but dedicated fuzzy chips are now available. For example, Adaptive Logic INC of San Jose, California, sells a “fuzzy chip”, the AL220, that can accept four analog inputs and generate four analog outputs. A block diagram of the chip is shown below: +------+ +------+ analog -−4-->| analog | | mux / +-−4--> analog in | mux | | SH | out +----+----+ +------+ | ^ V | +------+ +--+--+ | ADC / latch | | DAC | +------+------+ +-----+ | ^ | | 8 +------+ | | | | V | | +------+ +------+ | +-->| fuzzifier | | defuzzifier +--+ +-----+-----+ +------+ | ^ | +------+ | | | rule | | +->| processor +--+ | (50 rules) | +------+------+ | +------+------+ | parameter | | memory | | 256 x 8 | +------+ ADC: analog-to-digital converter DAC: digital-to-analog converter SH: sample/hold
14.4 Antilock brakes
As a first example, consider an anti-lock braking system, directed by a microcontroller chip. The microcontroller has to make decisions based on brake temperature, speed, and other variables in the system. The variable “temperature” in this system can be subdivided into a range of “states": “cold”, “cool”, “moderate”, “warm”, “hot”, “very hot”. The transition from one state to the next is hard to define. An arbitrary static threshold might be set to divide “warm” from “hot”. For example, at exactly 90 degrees, warm ends and hot begins. But this would result in a discontinuous change when the input value passed over that threshold. The transition wouldn't be smooth, as would be required in braking situations. The way around this is to make the states fuzzy. That is, allow them to change gradually from one state to the next. In order to do this there must be a dynamic relationship established between different factors. We start by defining the input temperature states using “membership functions": 14.5. LOGICAL INTERPRETATION OF FUZZY CONTROL 39
With this scheme, the input variable’s state no longer jumps abruptly from one state to the next. Instead, as the temperature changes, it loses value in one membership function while gaining value in the next. In other words, its ranking in the category of cold decreases as it becomes more highly ranked in the warmer category. At any sampled timeframe, the “truth value” of the brake temperature will almost always be in some degree part of two membership functions: i.e.: '0.6 nominal and 0.4 warm', or '0.7 nominal and 0.3 cool', and so on. The above example demonstrates a simple application, using the abstraction of values from multiple values. This only represents one kind of data, however, in this case, temperature. Adding additional sophistication to this braking system, could be done by additional factors such as traction, speed, inertia, set up in dynamic functions, according to the designed fuzzy system.[1]
14.5 Logical interpretation of fuzzy control
In spite of the appearance there are several difficulties to give a rigorous logical interpretation of the IF-THEN rules. As an example, interpret a rule as IF (temperature is “cold”) THEN (heater is “high”) by the first order for- mula Cold(x)→High(y) and assume that r is an input such that Cold(r) is false. Then the formula Cold(r)→High(t) is true for any t and therefore any t gives a correct control given r. A rigorous logical justification of fuzzy control is given in Hájek’s book (see Chapter 7) where fuzzy control is represented as a theory of Hájek’s basic logic. Also in Gerla 2005 a logical approach to fuzzy control is proposed based on fuzzy logic programming. Indeed, denote by f the fuzzy function arising of an IF-THEN systems of rules. Then we can translate this system into a fuzzy program P containing a series of rules whose head is “Good(x,y)". The interpretation of this predicate in the least fuzzy Herbrand model of P coincides with f. This gives further useful tools to fuzzy control.
14.6 See also
• Dynamic logic
• Bayesian inference
• Function approximation
• Fuzzy markup language
• Neural networks
• Neuro-fuzzy
• Fuzzy control language
• Type-2 fuzzy sets and systems 40 CHAPTER 14. FUZZY CONTROL SYSTEM
14.7 References
[1] Vichuzhanin, Vladimir (12 April 2012). “Realization of a fuzzy controller with fuzzy dynamic correction”. Central Euro- pean Journal of Engineering 2 (3): 392–398. doi:10.2478/s13531-012-0003-7.
• Gerla G., Fuzzy Logic Programming and fuzzy control, Studia Logica, 79 (2005) 231-254. • Bastian A., Identifying Fuzzy Models utilizing Genetic Programming, Fuzzy Sets and Systems 113, 333–350, 2000 • Hájek P., Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.
• Mamdani, E. H., Application of fuzzy algorithms for the control of a simple dynamic plant. In Proc IEEE (1974), 121-159.
14.8 Further reading
• Kevin M. Passino and Stephen Yurkovich, Fuzzy Control, Addison Wesley Longman, Menlo Park, CA, 1998 (522 pages)
• Kazuo Tanaka; Hua O. Wang (2001). Fuzzy control systems design and analysis: a linear matrix inequality approach. John Wiley and Sons. ISBN 978-0-471-32324-2.
• Cox, E. (Oct. 1992). Fuzzy fundamentals. Spectrum, IEEE, 29:10. pp. 58–61.
• Cox, E. (Feb. 1993) Adaptive fuzzy systems. Spectrum, IEEE, 30:2. pp. 7–31. • Jan Jantzen, “Tuning Of Fuzzy PID Controllers”, Technical University of Denmark, report 98-H 871, Septem- ber 30, 1998. • Jan Jantzen, Foundations of Fuzzy Control. Wiley, 2007 (209 pages) (Table of contents)
• Computational Intelligence: A Methodological Introduction by Kruse, Borgelt, Klawonn, Moewes, Steinbrecher, Held, 2013, Springer, ISBN 9781447150121
14.9 External links
• Introduction to Fuzzy Control • Fuzzy Logic in Embedded Microcomputers and Control Systems
• IEC 1131-7 CD1 IEC 1131-7 CD1 PDF • Online interactive demonstration of a system with 3 fuzzy rules Chapter 15
Fuzzy electronics
Fuzzy electronics is an electronic technology that uses fuzzy logic, instead of the two-state Boolean logic more commonly used in digital electronics. It has a wide range of applications, including control systems and artificial intelligence.
15.1 See also
• Defuzzification • Fuzzy set
• Fuzzy set operations
15.2 Bibliography
• Introduction to Applied Fuzzy Electronics, by Ahmad M. Ibrahim, ISBN 0-13-206400-6.
15.3 External links
• Applications of Fuzzy logic in electronics
41 Chapter 16
Fuzzy finite element
The fuzzy finite element method combines the well-established finite element method with the concept of fuzzy numbers, the latter being a special case of a fuzzy set.[1] The advantage of using fuzzy numbers instead of real numbers lies in the incorporation of uncertainty (on material properties, parameters, geometry, initial conditions, etc.) in the finite element analysis. One way to establish a fuzzy finite element (FE) analysis is to use existing FE software (in-house or commercial) as an inner-level module to compute a deterministic result, and to add an outer-level loop to handle the fuzziness (uncertainty). This outer-level loop comes down to solving an optimization problem. If the inner-level deterministic module produces monotonic behavior with respect to the input variables, then the outer-level optimization problem is greatly simplified, since in this case the extrema will be located at the vertices of the domain.
16.1 See also
• Finite element method
• Fuzzy number • Fuzzy set
• Uncertainty
16.2 References
[1] Michael Hanss, 2005. Applied Fuzzy Arithmetic, An Introduction with Engineering Applications. Springer, ISBN 3-540- 24201-5
42 Chapter 17
Fuzzy logic
Fuzzy logic is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1. By contrast, in Boolean logic, the truth values of variables may only be 0 or 1. Fuzzy logic has been extended to handle the concept of partial truth, where the truth value may range between completely true and completely false.[1] Furthermore, when linguistic variables are used, these degrees may be managed by specific functions.[2] The term “fuzzy logic” was introduced with the 1965 proposal of fuzzy set theory by Lotfi A. Zadeh.[3][4] Fuzzy logic has been applied to many fields, from control theory to artificial intelligence. Fuzzy logic had, however, been studied since the 1920s, as infinite-valued logic—notably by Łukasiewicz and Tarski.[5]
17.1 Overview
Classical logic only permits propositions having a value of truth or falsity. The notion of whether 1+1=2 is an absolute, immutable and mathematical truth. However, there exist certain propositions with variable answers, such as asking various people to identify a colour. The notion of truth doesn't fall by the wayside, but rather on a means of representing and reasoning over partial knowledge when afforded, by aggregating all possible outcomes into a dimensional spectrum. Both degrees of truth and probabilities range between 0 and 1 and hence may seem similar at first. For example, let a 100 ml glass contain 30 ml of water. Then we may consider two concepts: empty and full. The meaning of each of them can be represented by a certain fuzzy set. Then one might define the glass as being 0.7 empty and 0.3 full. Note that the concept of emptiness would be subjective and thus would depend on the observer or designer. Another designer might, equally well, design a set membership function where the glass would be considered full for all values down to 50 ml. It is essential to realize that fuzzy logic uses truth degrees as a mathematical model of the vagueness phenomenon while probability is a mathematical model of ignorance.
17.1.1 Applying truth values
A basic application might characterize various sub-ranges of a continuous variable. For instance, a temperature measurement for anti-lock brakes might have several separate membership functions defining particular temperature ranges needed to control the brakes properly. Each function maps the same temperature value to a truth value in the 0 to 1 range. These truth values can then be used to determine how the brakes should be controlled. In this image, the meanings of the expressions cold, warm, and hot are represented by functions mapping a temperature scale. A point on that scale has three “truth values” — one for each of the three functions. The vertical line in the image represents a particular temperature that the three arrows (truth values) gauge. Since the red arrow points to zero, this temperature may be interpreted as “not hot”. The orange arrow (pointing at 0.2) may describe it as “slightly warm” and the blue arrow (pointing at 0.8) “fairly cold”.
43 44 CHAPTER 17. FUZZY LOGIC
cold warm hot 1
0 temperature
Fuzzy logic temperature
17.1.2 Linguistic variables
While variables in mathematics usually take numerical values, in fuzzy logic applications, the non-numeric are often used to facilitate the expression of rules and facts.[6] A linguistic variable such as age may have a value such as young or its antonym old. However, the great utility of linguistic variables is that they can be modified via linguistic hedges applied to primary terms. These linguistic hedges can be associated with certain functions.
17.2 Early applications
The Japanese were the first to utilize fuzzy logic for practical applications. The first notable application was on the high-speed train in Sendai, in which fuzzy logic was able to improve the economy, comfort, and precision of the ride.[7] It has also been used in recognition of hand written symbols in Sony pocket computers; flight aid for helicopters; controlling of subway systems in order to improve driving comfort, precision of halting, and power economy; improved fuel consumption for auto mobiles; single-button control for washing machines, automatic motor control for vacuum cleaners with recognition of surface condition and degree of soiling; and prediction systems for early recognition of earthquakes through the Institute of Seismology Bureau of Metrology, Japan.[8]
17.3 Example
17.3.1 Hard science with IF-THEN rules
Fuzzy set theory defines fuzzy operators on fuzzy sets. The problem in applying this is that the appropriate fuzzy operator may not be known. For example, a simple temperature regulator that uses a fan might look like this: IF temperature IS very cold THEN stop fan IF temperature IS cold THEN turn down fan IF temperature IS normal THEN maintain fan IF temperature IS hot THEN speed up fan
There is no “ELSE” – all of the rules are evaluated, because the temperature might be “cold” and “normal” at the same time to different degrees. The AND, OR, and NOT operators of boolean logic exist in fuzzy logic, usually defined as the minimum, maximum, and complement; when they are defined this way, they are called the Zadeh operators. So for the fuzzy variables x and y: NOT x = (1 - truth(x)) x AND y = minimum(truth(x), truth(y)) x OR y = maximum(truth(x), truth(y)) 17.4. LOGICAL ANALYSIS 45
There are also other operators, more linguistic in nature, called hedges that can be applied. These are generally adverbs such as “very”, or “somewhat”, which modify the meaning of a set using a mathematical formula.
17.3.2 Define with multiply
x AND y = x*y x OR y = 1-(1-x)*(1-y)
1-(1-x)*(1-y) comes from this: x OR y = NOT( AND( NOT(x), NOT(y) ) ) x OR y = NOT( AND(1-x, 1-y) ) x OR y = NOT( (1-x)*(1-y) ) x OR y = 1-(1-x)*(1-y)
17.3.3 Define with sigmoid
sigmoid(x)=1/(1+e^-x) sigmoid(x)+sigmoid(-x) = 1 (sigmoid(x)+sigmoid(-x))*(sigmoid(y)+sigmoid(-y))*(sigmoid(z)+sigmoid(- z)) = 1
17.4 Logical analysis
In mathematical logic, there are several formal systems of “fuzzy logic"; most of them belong among so-called t-norm fuzzy logic.
17.4.1 Propositional fuzzy logics
The most important propositional fuzzy logics are:-
• Monoidal t-norm-based propositional fuzzy logic MTL is an axiomatization of logic where conjunction is defined by a left continuous t-norm and implication is defined as the residuum of the t-norm. Its models correspond to MTL-algebras that are pre-linear commutative bounded integral residuated lattices. • Basic propositional fuzzy logic BL is an extension of MTL logic where conjunction is defined by a continuous t-norm, and implication is also defined as the residuum of the t-norm. Its models correspond to BL-algebras. • Łukasiewicz fuzzy logic is the extension of basic fuzzy logic BL where standard conjunction is the Łukasiewicz t-norm. It has the axioms of basic fuzzy logic plus an axiom of double negation, and its models correspond to MV-algebras. • Gödel fuzzy logic is the extension of basic fuzzy logic BL where conjunction is Gödel t-norm. It has the axioms of BL plus an axiom of idempotence of conjunction, and its models are called G-algebras. • Product fuzzy logic is the extension of basic fuzzy logic BL where conjunction is product t-norm. It has the axioms of BL plus another axiom for cancellativity of conjunction, and its models are called product algebras. • Fuzzy logic with evaluated syntax (sometimes also called Pavelka’s logic), denoted by EVŁ, is a further gen- eralization of mathematical fuzzy logic. While the above kinds of fuzzy logic have traditional syntax and many-valued semantics, in EVŁ is evaluated also syntax. This means that each formula has an evaluation. Ax- iomatization of EVŁ stems from Łukasziewicz fuzzy logic. A generalization of classical Gödel completeness theorem is provable in EVŁ.
17.4.2 Predicate fuzzy logics
These extend the above-mentioned fuzzy logics by adding universal and existential quantifiers in a manner similar to the way that predicate logic is created from propositional logic. The semantics of the universal (resp. existential) quantifier in t-norm fuzzy logics is the infimum (resp. supremum) of the truth degrees of the instances of the quantified subformula. 46 CHAPTER 17. FUZZY LOGIC
17.4.3 Decidability issues for fuzzy logic
The notions of a “decidable subset” and "recursively enumerable subset” are basic ones for classical mathematics and classical logic. Thus the question of a suitable extension of these concepts to fuzzy set theory arises. A first proposal in such a direction was made by E.S. Santos by the notions of fuzzy Turing machine, Markov normal fuzzy algorithm and fuzzy program (see Santos 1970). Successively, L. Biacino and G. Gerla argued that the proposed definitions are rather questionable and therefore they proposed the following ones. Denote by Ü the set of rational numbers in [0,1]. Then a fuzzy subset s : S → [0,1] of a set S is recursively enumerable if a recursive map h : S×N → Ü exists such that, for every x in S, the function h(x,n) is increasing with respect to n and s(x) = lim h(x,n). We say that s is decidable if both s and its complement –s are recursively enumerable. An extension of such a theory to the general case of the L-subsets is possible (see Gerla 2006). The proposed definitions are well related with fuzzy logic. Indeed, the following theorem holds true (provided that the deduction apparatus of the considered fuzzy logic satisfies some obvious effectiveness property). Theorem. Any axiomatizable fuzzy theory is recursively enumerable. In particular, the fuzzy set of logically true formulas is recursively enumerable in spite of the fact that the crisp set of valid formulas is not recursively enumerable, in general. Moreover, any axiomatizable and complete theory is decidable. It is an open question to give supports for a Church thesis for fuzzy mathematics the proposed notion of recursive enumerability for fuzzy subsets is the adequate one. To this aim, an extension of the notions of fuzzy grammar and fuzzy Turing machine should be necessary (see for example Wiedermann’s paper). Another open question is to start from this notion to find an extension of Gödel's theorems to fuzzy logic. It is known that any boolean logic function could be represented using a truth table mapping each set of variable values into set of values {0, 1} . The task of synthesis of boolean logic function given in tabular form is one of basic tasks in traditional logic that is solved via disjunctive (conjunctive) perfect normal form. Each fuzzy (continuous) logic function could be represented by a choice table containing all possible variants of comparing arguments and their negations. A choice table maps each variant into value of an argument or a negation of an argument. For instance, for two arguments a row of choice table contains a variant of comparing values x1 , ¬x1 , x2 , ¬x2 and the corresponding function value f(x2 ≤ ¬x1 ≤ x1 ≤ ¬x2) = ¬x1 . The task of synthesis of fuzzy logic function given in tabular form was solved in.[9] New concepts of constituents of minimum and maximum were introduced. The sufficient and necessary conditions that a choice table defines a fuzzy logic function were derived.
17.5 Fuzzy databases
Once fuzzy relations are defined, it is possible to develop fuzzy relational databases. The first fuzzy relational database, FRDB, appeared in Maria Zemankova’s dissertation. Later, some other models arose like the Buckles-Petry model, the Prade-Testemale Model, the Umano-Fukami model or the GEFRED model by J.M. Medina, M.A. Vila et al. In the context of fuzzy databases, some fuzzy querying languages have been defined, highlighting the SQLf by P. Bosc et al. and the FSQL by J. Galindo et al. These languages define some structures in order to include fuzzy aspects in the SQL statements, like fuzzy conditions, fuzzy comparators, fuzzy constants, fuzzy constraints, fuzzy thresholds, linguistic labels and so on. Much progress has been made to take fuzzy logic database applications to the web and let the world easily use them, for example: http://sullivansoftwaresystems.com/cgi-bin/fuzzy-logic-match-algorithm.cgi?SearchString=garia This en- ables fuzzy logic matching to be incorporated into a database system or application.
17.6 Comparison to probability
Fuzzy logic and probability address different forms of uncertainty. While both fuzzy logic and probability theory can represent degrees of certain kinds of subjective belief, fuzzy set theory uses the concept of fuzzy set membership, i.e., how much a variable is in a set (there is not necessarily any uncertainty about this degree), and probability theory uses the concept of subjective probability, i.e., how probable is it that a variable is in a set (it either entirely is or entirely is not in the set in reality, but there is uncertainty around whether it is or is not). The technical consequence of this distinction is that fuzzy set theory relaxes the axioms of classical probability, which are themselves derived from adding uncertainty, but not degree, to the crisp true/false distinctions of classical Aristotelian logic. 17.7. RELATION TO ECORITHMS 47
Bruno de Finetti argues that only one kind of mathematical uncertainty, probability, is needed, and thus fuzzy logic is unnecessary. However, Bart Kosko shows in Fuzziness vs. Probability that probability theory is a subtheory of fuzzy logic, as questions of degrees of belief in mutually-exclusive set membership in probability theory can be represented as certain cases of non-mutually-exclusive graded membership in fuzzy theory. In that context, he also derives Bayes’ theorem from the concept of fuzzy subsethood. Lotfi A. Zadeh argues that fuzzy logic is different in character from probability, and is not a replacement for it. He fuzzified probability to fuzzy probability and also generalized it to possibility theory. (cf.[10]) More generally, fuzzy logic is one of many different extensions to classical logic intended to deal with issues of uncertainty outside of the scope of classical logic, the inapplicability of probability theory in many domains, and the paradoxes of Dempster-Shafer theory. See also probabilistic logics.
17.7 Relation to ecorithms
Leslie Valiant, a winner of the Turing Award, uses the term “ecorithms” to describe how many less exact systems and techniques like fuzzy logic (and “less robust” logic) can be applied to learning algorithms. Valiant essentially redefines machine learning as evolutionary. Ecorithms and fuzzy logic also have the common property of dealing with possibilities more than probabilities, although feedback and feed forward, basically stochastic “weights,” are a feature of both when dealing with, for example, dynamical systems. In general use, ecorithms are algorithms that learn from their more complex environments (Hence Eco) to generalize, approximate and simplify solution logic. Like fuzzy logic, they are methods used to overcome continuous variables or systems too complex to completely enumerate or understand discretely or exactly. See in particular p. 58 of the reference comparing induction/invariance, robust, mathematical and other logical limits in computing, where techniques including fuzzy logic and natural data selection (à la “computational Darwinism”) can be used to short-cut computational complexity and limits in a “practical” way (such as the brake temperature example in this article).[11]
17.8 Compensatory fuzzy logic
Compensatory fuzzy logic (CFL) is a branch of fuzzy logic with modified rules for conjunction and disjunction. When the truth value of one component of a conjunction or disjunction is increased or decreased, the other component is decreased or increased to compensate. This increase or decrease in truth value may be offset by the increase or decrease in another component. An offset may be blocked when certain thresholds are met. Proponents claim that CFL allows better semantic behavior. Compensatory Fuzzy Logic consists of four continuous operators: conjunction (c); disjunction (d); fuzzy strict order (or); and negation (n). The conjunction is the geometric mean and its dual as conjunctive and disjunctive operators.[12]
17.9 See also
• Adaptive neuro fuzzy inference system (ANFIS) • Artificial neural network • Defuzzification • Expert system • False dilemma • Fuzzy architectural spatial analysis • Fuzzy classification • Fuzzy concept • Fuzzy Control Language • Fuzzy control system 48 CHAPTER 17. FUZZY LOGIC
• Fuzzy electronics
• Fuzzy subalgebra
• FuzzyCLIPS
• High Performance Fuzzy Computing
• IEEE Transactions on Fuzzy Systems
• Interval finite element
• Machine learning
• Neuro-fuzzy
• Noise-based logic
• Rough set
• Sorites paradox
• Type-2 fuzzy sets and systems
• Vector logic
17.10 References
[1] Novák, V., Perfilieva, I. and Močkoř, J. (1999) Mathematical principles of fuzzy logic Dodrecht: Kluwer Academic. ISBN 0-7923-8595-0
[2] Ahlawat, Nishant, Ashu Gautam, and Nidhi Sharma (International Research Publications House 2014) “Use of Logic Gates to Make Edge Avoider Robot.” International Journal of Information & Computation Technology (Volume 4, Issue 6; page 630) ISSN 0974-2239 (Retrieved 27 April 2014)
[3] “Fuzzy Logic”. Stanford Encyclopedia of Philosophy. Stanford University. 2006-07-23. Retrieved 2008-09-30.
[4] Zadeh, L.A. (1965). “Fuzzy sets”. Information and Control 8 (3): 338–353. doi:10.1016/s0019-9958(65)90241-x.
[5] Pelletier, Francis Jeffry (2000). “Review of Metamathematics of fuzzy logics" (PDF). The Bulletin of Symbolic Logic 6 (3): 342–346. JSTOR 421060.
[6] Zadeh, L. A. et al. 1996 Fuzzy Sets, Fuzzy Logic, Fuzzy Systems, World Scientific Press, ISBN 981-02-2421-4
[7] Kosko, B (June 1, 1994). “Fuzzy Thinking: The New Science of Fuzzy Logic”. Hyperion.
[8] Bansod, Nitin A., Marshall Kulkarni, and S.H. Patil (Bharati Vidyapeeth College of Engineering) “Soft Computing- A Fuzzy Logic Approach”. Soft Computing (Allied Publishers 2005) (page 73)
[9] Zaitsev D.A., Sarbei V.G., Sleptsov A.I., Synthesis of continuous-valued logic functions defined in tabular form, Cyber- netics and Systems Analysis, Volume 34, Number 2 (1998), 190-195.
[10] Novák, V (2005). “Are fuzzy sets a reasonable tool for modeling vague phenomena?". Fuzzy Sets and Systems 156: 341– 348. doi:10.1016/j.fss.2005.05.029.
[11] Valiant, Leslie, (2013) Probably Approximately Correct: Nature’s Algorithms for Learning and Prospering in a Complex World New York: Basic Books. ISBN 978-0465032716
[12] Cejas, Jesús, (2011) Compensatory Fuzzy Logic. La Habana: Revista de Ingeniería Industrial. ISSN 1815-5936 17.11. BIBLIOGRAPHY 49
17.11 Bibliography
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• Biacino, L.; Gerla, G. (2002). “Fuzzy logic, continuity and effectiveness”. Archive for Mathematical Logic 41 (7): 643–667. doi:10.1007/s001530100128. ISSN 0933-5846.
• Cox, Earl (1994). The fuzzy systems handbook: a practitioner’s guide to building, using, maintaining fuzzy systems. Boston: AP Professional. ISBN 0-12-194270-8.
• Gerla, Giangiacomo (2006). “Effectiveness and Multivalued Logics”. Journal of Symbolic Logic 71 (1): 137– 162. doi:10.2178/jsl/1140641166. ISSN 0022-4812.
• Hájek, Petr (1998). Metamathematics of fuzzy logic. Dordrecht: Kluwer. ISBN 0-7923-5238-6.
• Hájek, Petr (1995). “Fuzzy logic and arithmetical hierarchy”. Fuzzy Sets and Systems 3 (8): 359–363. doi:10.1016/0165-0114(94)00299-M. ISSN 0165-0114.
• Halpern, Joseph Y. (2003). Reasoning about uncertainty. Cambridge, Mass: MIT Press. ISBN 0-262-08320- 5.
• Höppner, Frank; Klawonn, F.; Kruse, R.; Runkler, T. (1999). Fuzzy cluster analysis: methods for classification, data analysis and image recognition. New York: John Wiley. ISBN 0-471-98864-2.
• Ibrahim, Ahmad M. (1997). Introduction to Applied Fuzzy Electronics. Englewood Cliffs, N.J: Prentice Hall. ISBN 0-13-206400-6.
• Klir, George J.; Folger, Tina A. (1988). Fuzzy sets, uncertainty, and information. Englewood Cliffs, N.J: Prentice Hall. ISBN 0-13-345984-5.
• Klir, George J.; St Clair, Ute H.; Yuan, Bo (1997). Fuzzy set theory: foundations and applications. Englewood Cliffs, NJ: Prentice Hall. ISBN 0-13-341058-7.
• Klir, George J.; Yuan, Bo (1995). Fuzzy sets and fuzzy logic: theory and applications. Upper Saddle River, NJ: Prentice Hall PTR. ISBN 0-13-101171-5.
• Kosko, Bart (1993). Fuzzy thinking: the new science of fuzzy logic. New York: Hyperion. ISBN 0-7868-8021- X.
• Kosko, Bart; Isaka, Satoru (July 1993). “Fuzzy Logic”. Scientific American 269 (1): 76–81. doi:10.1038/scientificamerican0793- 76.
• Lohani, A.K.; Goel, N.K.; Bhatia K.K.S. (2006). “Takagi–Sugeno fuzzy inference system for modeling stage– discharge relationship”. Journal of Hydrology 331 (1): 146–160. doi:10.1016/j.jhydrol.2006.05.007.
• Lohani, A.K.; Goel, N.K.; Bhatia K.K.S. (2007). “Deriving stage–discharge–sediment concentration relation- ships using fuzzy logic”. Hydrological Sciences Journal 52 (4): 793–807. doi:10.1623/hysj.52.4.793.
• Lohani, A.K.; Goel, N.K.; Bhatia K.K.S. (2011). “Comparative study of neural network, fuzzy logic and linear transfer function techniques in daily rainfall‐runoff modelling under different input domains”. Hydrological Processes 25 (2): 175–193. doi:10.1002/hyp.7831.
• Lohani, A.K.; Goel, N.K.; Bhatia K.K.S. (2012). “Hydrological time series modeling: A comparison between adaptive neuro-fuzzy, neural network and autoregressive techniques”. Journal of Hydrology. 442-443 (6): 23–35. doi:10.1016/j.jhydrol.2012.03.031.
• Malek Masmoudi and Alain Haït, Project scheduling under uncertainty using fuzzy modeling and solving tech- niques, Engineering Applications of Artificial Intelligence - Elsevier, July 2012.
• Malek Masmoudi and Alain Haït, Fuzzy uncertainty modelling for project planning; application to helicopter maintenance, International Journal of Production Research, Vol 50, issue 24, November2012.
• Montagna, F. (2001). “Three complexity problems in quantified fuzzy logic”. Studia Logica 68 (1): 143–152. doi:10.1023/A:1011958407631. ISSN 0039-3215. 50 CHAPTER 17. FUZZY LOGIC
• Mundici, Daniele; Cignoli, Roberto; D'Ottaviano, Itala M. L. (1999). Algebraic foundations of many-valued reasoning. Dodrecht: Kluwer Academic. ISBN 0-7923-6009-5.
• Novák, Vilém (1989). Fuzzy Sets and Their Applications. Bristol: Adam Hilger. ISBN 0-85274-583-4.
• Novák, Vilém (2005). “On fuzzy type theory”. Fuzzy Sets and Systems 149 (2): 235–273. doi:10.1016/j.fss.2004.03.027.
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• Onses, Richard (1994). Détermination de l´incertitude inhérente aux investissements en Amérique Latine sur la base de la théorie des sous ensembles flous. Barcelona. ISBN 84-475-0881-1.
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• Pedrycz, Witold; Gomide, Fernando (2007). Fuzzy systems engineering: Toward Human-Centerd Computing. Hoboken: Wiley-Interscience. ISBN 978-0-471-78857-7.
• Pu, Pao Ming; Liu, Ying Ming (1980). “Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore- Smith convergence”. Journal of Mathematical Analysis and Applications 76 (2): 571–599. doi:10.1016/0022- 247X(80)90048-7. ISSN 0022-247X.
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• Santos, Eugene S. (1970). “Fuzzy Algorithms”. Information and Control 17 (4): 326–339. doi:10.1016/S0019- 9958(70)80032-8.
• Scarpellini, Bruno (1962). “Die Nichaxiomatisierbarkeit des unendlichwertigen Prädikatenkalküls von Łukasiewicz”. Journal of Symbolic Logic (Association for Symbolic Logic) 27 (2): 159–170. doi:10.2307/2964111. ISSN 0022-4812. JSTOR 2964111.
• Seising, Rudolf (2007). The Fuzzification of Systems. The Genesis of Fuzzy Set Theory and Its Initial Applica- tions -- Developments up to the 1970s. Springer-Verlag. ISBN 978-3-540-71795-9.
• Steeb, Willi-Hans (2008). The Nonlinear Workbook: Chaos, Fractals, Cellular Automata, Neural Networks, Genetic Algorithms, Gene Expression Programming, Support Vector Machine, Wavelets, Hidden Markov Models, Fuzzy Logic with C++, Java and SymbolicC++ Programs: 4edition. World Scientific. ISBN 981-281-852-9.
• Tsitolovsky, Lev; Sandler, Uziel (2008). Neural Cell Behavior and Fuzzy Logic. Springer. ISBN 978-0-387- 09542-4.
• Wiedermann, J. (2004). “Characterizing the super-Turing computing power and efficiency of classical fuzzy Turing machines”. Theor. Comput. Sci. 317 (1–3): 61–69. doi:10.1016/j.tcs.2003.12.004.
• Yager, Ronald R.; Filev, Dimitar P. (1994). Essentials of fuzzy modeling and control. New York: Wiley. ISBN 0-471-01761-2.
• Van Pelt, Miles (2008). Fuzzy Logic Applied to Daily Life. Seattle, WA: No No No No Press. ISBN 0-252- 16341-9.
• Von Altrock, Constantin (1995). Fuzzy logic and NeuroFuzzy applications explained. Upper Saddle River, NJ: Prentice Hall PTR. ISBN 0-13-368465-2.
• Wilkinson, R.H. (1963). “A method of generating functions of several variables using analog diode logic”. IEEE Transactions on Electronic Computers 12 (2): 112–129. doi:10.1109/PGEC.1963.263419.
• Zadeh, L.A. (1968). “Fuzzy algorithms”. Information and Control 12 (2): 94–102. doi:10.1016/S0019- 9958(68)90211-8. ISSN 0019-9958. 17.12. EXTERNAL LINKS 51
• Zadeh, L.A. (1965). “Fuzzy sets”. Information and Control 8 (3): 338–353. doi:10.1016/S0019-9958(65)90241- X. ISSN 0019-9958. • Zemankova-Leech, M. (1983). “Fuzzy Relational Data Bases”. Ph. D. Dissertation. Florida State University.
• Zimmermann, H. (2001). Fuzzy set theory and its applications. Boston: Kluwer Academic Publishers. ISBN 0-7923-7435-5.
• Moghaddam, M. J., M. R. Soleymani, and M. A. Farsi. “Sequence planning for stamping operations in pro- gressive dies.” Journal of Intelligent Manufacturing(2013): 1-11.
17.12 External links
• Formal fuzzy logic - article at Citizendium
• Fuzzy Logic - article at Scholarpedia • Modeling With Words - article at Scholarpedia
• Fuzzy logic - article at Stanford Encyclopedia of Philosophy • Fuzzy Math - Beginner level introduction to Fuzzy Logic
• Fuzzylite - A cross-platform, free open-source Fuzzy Logic Control Library written in C++. Also has a very useful graphic user interface in QT4.
• Online Calculator based upon Fuzzy logic – Gives online calculation in educational example of fuzzy logic model. Chapter 18
Fuzzy markup language
Fuzzy Markup Language (FML) is a specific purpose markup language based on XML, used for describing the structure and behavior of a fuzzy system independently of the hardware architecture devoted to host and run it.
18.1 Overview
FML was designed and developed by Giovanni Acampora during his Ph.D. course in Computer Science, under the supervision of Prof. Vincenzo Loia, at University of Salerno, Italy, in 2004. The original idea inspired Giovanni Acampora to create FML was the necessity of creating a cooperative fuzzy-based framework aimed at automatically controlling a living environment characterized by a plethora of heterogeneous devices whose interactions were devoted to maximize the human comfort under energy saving constraints. This framework represented one of the first concrete examples of Ambient Intelligence. Beyond this pioneering application, the major advantage of using XML to describe a fuzzy system is hardware/software interoperability. Indeed, all that is needed to read an FML file is the appropriate schema for that file, and an FML parser. This markup approach makes it much easier to exchange fuzzy systems between software: for example, a machine learning application could extract fuzzy rules which could then be read directly into a fuzzy inference engine or uploaded into a fuzzy controller. Also, with technologies like XSLT, it is possible to compile the FML into the programming language of your choice, ready for embedding into whatever application you please. As stated by Mike Watts on his popular Computational Intelligence blog:[1]
“Although Acampora’s motivation for developing FML seems to be to develop embedded fuzzy controllers for ambient intelligence applications, FML could be a real boon for developers of fuzzy rule extraction algorithms: from my own experience during my PhD, I know that having to design a file format and implement the appropriate parsers for rule extraction and fuzzy inference engines can be a real pain, taking as much time as implementing the rule extraction algorithm itself. I would much rather have used something like FML for my work.”
A complete overview of FML and related applications can be found in the book titled On the power of Fuzzy Markup Language[2] edited by Giovanni Acampora, Chang-Shing Lee, Vincenzo Loia and Mei-Hui Wang, and published by Springer in the series Studies on Fuzziness and Soft Computing.
18.2 FML at work: syntax, grammar and hardware synthesis
FML allows fuzzy systems to be coded through a collection of correlated semantic tags capable of modeling the different components of a classical fuzzy controller such as knowledge base, rule base, fuzzy variables and fuzzy rules. Therefore, the FML tags used to build a fuzzy controller represent the set of lexemes used to create fuzzy expressions. In order to design a well-formed XML-based language, an FML context-free grammar is defined by means of a XML schema which defines name, type and attributes characterized each XML element. However, since an FML program represents only a static view of a fuzzy logic controller, the so-called eXtensible Stylesheet Language Translator (XSLT) is provided to change this static view to a computable version. Indeed, XSLTs modules are able to convert the FML-based fuzzy controller in a general purpose computer language using an XSL file containing the
52 18.2. FML AT WORK: SYNTAX, GRAMMAR AND HARDWARE SYNTHESIS 53
translation description. At this level, the control is executable for the hardware. In short, FML is essentially composed by three layers:
• XML in order to create a new markup language for fuzzy logic control;
• a XML Schema in order to define the legal building blocks;
• eXtensible Stylesheet Language Transformations (XSLT) in order to convert a fuzzy controller description into a specific programming language.
18.2.1 FML Syntax
FML syntax is composed of XML tags and attributes which describe the different components of a fuzzy logic controller listed below:
• fuzzy knowledge base;
• fuzzy rule base;
• inference engine
• fuzzification subsystem;
• defuzzification subsystem.
In detail, the opening tag of each FML program is
•
•
• a set of tags defining a shape of fuzzy sets are related to fuzzy terms.
The attributes of
•
•
•
•
•
•
Every shaping tag uses a set of attributes which defines the real outline of corresponding fuzzy set. The number of these attributes depends on the chosen fuzzy set shape. In order to make an example, let us consider the Tipper Inference System described in Mathwork Matlab Fuzzy Logic Toolbox Tutorial. This Mamdani system is used to regulate the tipping in, for example, a restaurant. It has got two variables in input (food and service) and one in output (tip). FML code for modeling part of knowledge base of this fuzzy system containing variables food and tip is shown below.
A special tag that can furthermore be used to define a fuzzy shape is
Now, let us see a Takagi-Sugeno-Kang system that regulates the same issue. The most important difference with Mamdani system is the definition of a different output variable tip. The
The FML definition of rule base component in a Tsk system doesn’t change a lot. The only different thing is that the
18.2.2 FML Grammar
The FML tags used to build a fuzzy controller represent the set of lexemes used to create fuzzy expressions. However, in order to realize a well-formed XML-based language, an FML context-free grammar is necessary and described in the following. The FML context-free grammar is modeled by XML file in the form of a XML Schema Document (XSD) which expresses the set of rules to which a document must conform in order to be considered a valid FML document. Based on the previous definition, a portion of the FML XSD regarding the knowledge base definition is given below.
18.2.3 FML Synthesis
Since an FML program realizes only a static view of a fuzzy system, the so-called eXtensible Stylesheet Language Translator (XSLT) is provided to change this static view to a computable version. In particular, the XSLT technology is used convert a fuzzy controller description into a general-purpose computer language to be computed on several hardware platforms. Currently, a XSLT converting FML program in runnable Java code has been implemented. In this way, thanks to the transparency capabilities provided by Java virtual machines, it is possible to obtain a fuzzy controller modeled in high level way by means of FML and runnable on a plethora of hardware architectures through Java technologies. However, XSLT can be also used for converting FML programs in legacy languages related to a particular hardware or in other general purpose languages.
18.3 References
[1] Watts, Mike (2011-05-28). “Computational Intelligence: Fuzzy Markup Language”. Computational-intelligence.blogspot.it. Retrieved 2012-06-11.
[2] Acampora, Giovanni; Loia, Vincenzo; Lee, Chang-Shing; Wang, Mei-Hui, eds. (2013). On the power of Fuzzy Markup Language. Vol.296. Studies in Fuzziness and Soft Computing (Springer). Retrieved January 11, 2013. 18.4. FURTHER READING 57
18.4 Further reading
• Acampora, Giovanni and Loia, Vincenzo (2005). “Using FML and Fuzzy Technology in Adaptive Ambient Intelligence Environments” (PDF). Vol.1, No.2. International Journal of Computational Intelligence Research. pp. 171–182. Retrieved June 3, 2012.
• Lee, Chang-Shing et al. (December 2010). “Diet assessment based on type-2 fuzzy ontology and fuzzy markup language”. Volume 25, Issue 12. International Journal of Intelligent Systems. pp. 1187–1216. Retrieved June 3, 2012. (subscription required)
• Acampora, G.; Loia, V. (2005). “Fuzzy control interoperability and scalability for adaptive domotic frame- work”. IEEE Transactions on Industrial Informatics 1 (2): 97–111. doi:10.1109/TII.2005.844431.
• Acampora, G.; Loia, V. (2008). “A proposal of ubiquitous fuzzy computing for Ambient Intelligence”. Infor- mation Sciences 178 (3): 631–646. doi:10.1016/j.ins.2007.08.023.
• Acampora, G.; Wang, M.-H.; Lee, C.-S.; Hsieh, K.-L.; Hsu, C.-Y.; Chang, C.-C. (2010). “Ontology-based multi-agents for intelligent healthcare applications”. Journal of Ambient Intelligence and Humanized Computing 1 (2): 111–131. doi:10.1007/s12652-010-0011-5. • Acampora, G.; Loia, V.; Gaeta, M.; Vasilakos, A.V. (2010). “Interoperable and adaptive fuzzy services for ambient intelligence applications”. ACM Trans. Auton. Adapt. Syst. 5 (2). doi:10.1145/1740600.1740604. Chapter 19
Fuzzy mathematics
For other uses, see Fuzzy math (disambiguation).
Fuzzy mathematics forms a branch of mathematics related to fuzzy set theory and fuzzy logic. It started in 1965 after the publication of Lotfi Asker Zadeh's seminal work Fuzzy sets.[1] A fuzzy subset A of a set X is a function A:X→L, where L is the interval [0,1]. This function is also called a membership function. A membership function is a generalization of a characteristic function or an indicator function of a subset defined for L = {0,1}. More generally, one can use a complete lattice L in a definition of a fuzzy subset A .[2] The evolution of the fuzzification of mathematical concepts can be broken down into three stages:[3]
1. straightforward fuzzification during the sixties and seventies, 2. the explosion of the possible choices in the generalization process during the eighties, 3. the standardization, axiomatization and L-fuzzification in the nineties.
Usually, a fuzzification of mathematical concepts is based on a generalization of these concepts from characteristic functions to membership functions. Let A and B be two fuzzy subsets of X. Intersection A ∩ B and union A ∪ B are defined as follows: (A ∩ B)(x) = min(A(x),B(x)), (A ∪ B)(x) = max(A(x),B(x)) for all x ∈ X. Instead of min and max one can use t-norm and t-conorm, respectively ,[4] for example, min(a,b) can be replaced by multiplication ab. A straightforward fuzzification is usually based on min and max operations because in this case more properties of traditional mathematics can be extended to the fuzzy case. A very important generalization principle used in fuzzification of algebraic operations is a closure property. Let * be a binary operation on X. The closure property for a fuzzy subset A of X is that for all x,y ∈ X, A(x*y) ≥ min(A(x),A(y)). Let (G,*) be a group and A a fuzzy subset of G. Then A is a fuzzy subgroup of G if for all x,y in G, A(x*y−1) ≥ min(A(x),A(y−1)). A similar generalization principle is used, for example, for fuzzification of the transitivity property. Let R be a fuzzy relation in X, i.e. R is a fuzzy subset of X×X. Then R is transitive if for all x,y,z in X, R(x,z) ≥ min(R(x,y),R(y,z)).
19.1 Some fields of mathematics using fuzzy set theory
Fuzzy subgroupoids and fuzzy subgroups were introduced in 1971 by A. Rosenfeld .[5] Hundreds of papers on related topics have been published. Recent results and references can be found in [6] and.[7] Main results in fuzzy fields and fuzzy Galois theory are published in a 1998 paper.[8] Fuzzy topology was introduced by C.L. Chang[9] in 1968 and further was studied in many papers.[10] Main concepts of fuzzy geometry were introduced by Tim Poston in 1971,[11] A. Rosenfeld in 1974, by J.J. Buckley and E. Eslami in 1997[12] and by D. Ghosh and D. Chakraborty in 2012-14 [13] [14] Basic types of fuzzy relations were introduced by Zadeh in 1971.[15] The properties of fuzzy graphs have been studied by A. Kaufman,[16] A. Rosenfel,[17] and by R.T. Yeh and S.Y. Bang.[18] Recent results can be found in a 2000 article.[19]
58 19.2. SEE ALSO 59
Possibility theory, nonadditive measures, fuzzy measure theory and fuzzy integrals are studied in the cited articles and treatises.[20][21][22][23][24] Main results and references on formal fuzzy logic can be found in these citations.[25][26]
19.2 See also
• Fuzzy measure theory
• Fuzzy subalgebra
• Monoidal t-norm logic
• Possibility theory
• T-norm
19.3 References
[1] Zadeh, L. A. (1965) “Fuzzy sets”, Information and Control, 8, 338–353.
[2] Goguen, J. (1967) “L-fuzzy sets”, J. Math. Anal. Appl., 18, 145-174.
[3] Kerre, E.E., Mordeson, J.N. (2005) “A historical overview of fuzzy mathematics”, New Mathematics and Natural Compu- tation, 1, 1-26.
[4] Klement, E.P., Mesiar, R., Pap, E. (2000) Triangular Norms. Dordrecht, Kluwer.
[5] Rosenfeld, A. (1971) “Fuzzy groups”, J. Math. Anal. Appl., 35, 512-517.
[6] Mordeson, J.N., Malik, D.S., Kuroli, N. (2003) Fuzzy Semigroups. Studies in Fuzziness and Soft Computing, vol. 131, Springer-Verlag
[7] Mordeson, J.N., Bhutani, K.R., Rosenfeld, A. (2005) Fuzzy Group Theory. Studies in Fuzziness and Soft Computing, vol. 182. Springer-Verlag.
[8] Mordeson, J.N., Malik, D.S (1998) Fuzzy Commutative Algebra. World Scientific.
[9] Chang, C.L. (1968) “Fuzzy topological spaces”, J. Math. Anal. Appl., 24, 182—190.
[10] Liu, Y.-M., Luo, M.-K. (1997) Fuzzy Topology. Advances in Fuzzy Systems - Applications and Theory, vol. 9, World Scientific, Singapore.
[11] Poston, Tim, “Fuzzy Geometry”.
[12] Buckley, J.J., Eslami, E. (1997) “Fuzzy plane geometry I: Points and lines”. Fuzzy Sets and Systems, 86, 179-187.
[13] Ghosh, D., Chakraborty, D. (2012) “Analytical fuzzy plane geometry I”. Fuzzy Sets and Systems, 209, 66-83.
[14] Chakraborty, D. and Ghosh, D. (2014) “Analytical fuzzy plane geometry II”. Fuzzy Sets and Systems, 243, 84–109.
[15] Zadeh L.A. (1971) “Similarity relations and fuzzy orderings”. Inform. Sci., 3, 177–200.
[16] Kaufmann, A. (1973). Introduction a la théorie des sous-ensembles flous. Paris. Masson.
[17] A. Rosenfeld, A. (1975) “Fuzzy graphs”. In: Zadeh, L.A., Fu, K.S., Tanaka, K., Shimura, M. (eds.), Fuzzy Sets and their Applications to Cognitive and Decision Processes, Academic Press, New York, ISBN 978-0-12-775260-0, pp. 77–95.
[18] Yeh, R.T., Bang, S.Y. (1975) “Fuzzy graphs, fuzzy relations and their applications to cluster analysis”. In: Zadeh, L.A., Fu, K.S., Tanaka, K., Shimura, M. (eds.), Fuzzy Sets and their Applications to Cognitive and Decision Processes, Academic Press, New York, ISBN 978-0-12-775260-0, pp. 125–149.
[19] Mordeson, J.N., Nair, P.S. (2000) Fuzzy Graphs and Fuzzy Hypergraphs. Studies in Fuzziness and Soft Computing, vol. 46. Springer-Verlag.
[20] Zadeh, L.A. (1978) “Fuzzy sets as a basis for a theory of possibility”. Fuzzy Sets and Systems, 1, 3-28. 60 CHAPTER 19. FUZZY MATHEMATICS
[21] Dubois, D., Prade, H. (1988) Possibility Theory: An Approach to Computerized Processing of Uncertainty. Plenum Press, New York.
[22] Wang, Z., Klir, G.J. (1992) Fuzzy Measure Theory. Plenum Press.
[23] Klir, G.J. (2005) Uncertainty and Information. Foundations of Generalized Information Theory. Wiley.
[24] Sugeno, M. (1974) Theory of Fuzzy Integrals and its Applications. PhD Dissertation. Tokyo, Institute of Technology.
[25] Hájek, P. (1998) Metamathematics of Fuzzy Logic. Dordrecht: Kluwer.
[26] Esteva, F., Godo, L. (2001) “Monoidal t-norm based logic: Towards a logic of left-continuous t-norms”. Fuzzy Sets and Systems, 124, 271–288.
19.4 External links
• Zadeh, L.A. Fuzzy Logic - article at Scholarpedia
• Hajek, P. Fuzzy Logic - article at Stanford Encyclopedia of Philosophy • Navara, M. Triangular Norms and Conorms - article at Scholarpedia
• Dubois, D., Prade H. Possibility Theory - article at Scholarpedia • Center for Mathematics of Uncertainty Fuzzy Math Research - Web site hosted at Creighton University
• Seising, R. Book on the history of the mathematical theory of Fuzzy Sets: The Fuzzification of Systems. The Genesis of Fuzzy Set Theory and Its Initial Applications -- Developments up to the 1970s (Studies in Fuzziness and Soft Computing, Vol. 216) Berlin, New York, [et al.]: Springer 2007. Chapter 20
Fuzzy measure theory
In mathematics, fuzzy measure theory considers generalized measures in which the additive property is replaced by the weaker property of monotonicity. The central concept of fuzzy measure theory is the fuzzy measure (also capacity, see [1]) which was introduced by Choquet in 1953 and independently defined by Sugeno in 1974 in the context of fuzzy integrals. There exists a number of different classes of fuzzy measures including plausibility/belief measures; possibility/necessity measures; and probability measures which are a subset of classical measures.
20.1 Definitions
Let X be a universe of discourse, C be a class of subsets of X , and E,F ∈ C .A function g : C → R where
1. ∅ ∈ C ⇒ g(∅) = 0
2. E ⊆ F ⇒ g(E) ≤ g(F ) is called a fuzzy measure. A fuzzy measure is called normalized or regular if g(X) = 1 .
20.2 Properties of fuzzy measures
For any E,F ∈ C , a fuzzy measure is:
• additive if g(E ∪ F ) = g(E) + g(F ). for all E ∩ F = ∅ ;
• supermodular if g(E ∪ F ) + g(E ∩ F ) ≥ g(E) + g(F ) ;
• submodular if g(E ∪ F ) + g(E ∩ F ) ≤ g(E) + g(F ) ;
• superadditive if g(E ∪ F ) ≥ g(E) + g(F ) for all E ∩ F = ∅ ;
• subadditive if g(E ∪ F ) ≤ g(E) + g(F ) for all E ∩ F = ∅ ;
• symmetric if |E| = |F | implies g(E) = g(F ) ;
• Boolean if g(E) = 0 or g(E) = 1 .
Understanding the properties of fuzzy measures is useful in application. When a fuzzy measure is used to define a function such as the Sugeno integral or Choquet integral, these properties will be crucial in understanding the function’s behavior. For instance, the Choquet integral with respect to an additive fuzzy measure reduces to the Lebesgue integral. In discrete cases, a symmetric fuzzy measure will result in the ordered weighted averaging (OWA) operator. Submodular fuzzy measures result in convex functions, while supermodular fuzzy measures result in concave functions when used to define a Choquet integral.
61 62 CHAPTER 20. FUZZY MEASURE THEORY
20.3 Möbius representation
Let g be a fuzzy measure, the Möbius representation of g is given by the set function M, where for every E,F ⊆ X ,
∑ M(E) = (−1)|E\F |g(F ). F ⊆E The equivalent axioms in Möbius representation are:
1. M(∅) = 0 . ∑ ≥ ⊆ ∈ 2. F ⊆E|i∈F M(F ) 0 , for all E X and all i E ∑ A fuzzy measure in Möbius representation M is called normalized if E⊆X M(E) = 1. Möbius representation can be used to give an indication of which subsets of X interact with one another. For instance, an additive fuzzy measure has Möbius values all equal to zero except for singletons. The fuzzy measure g in standard representation can be recovered from the Möbius form using the Zeta transform:
∑ g(E) = M(F ), ∀E ⊆ X. F ⊆E
20.4 Simplification assumptions for fuzzy measures
Fuzzy measures are defined on a semiring of sets or monotone class which may be as granular as the power set of X, and even in discrete cases the number of variables can as large as 2|X|. For this reason, in the context of multi-criteria decision analysis and other disciplines, simplification assumptions on the fuzzy measure have been introduced so that it is less computationally expensive∑ to determine and use. For instance, when it is assumed the fuzzy measure is { } additive, it will hold that g(E) = i∈E g( i ) and the values of the fuzzy measure can be evaluated from the values on X. Similarly, a symmetric fuzzy measure is defined uniquely by |X| values. Two important fuzzy measures that can be used are the Sugeno- or λ -fuzzy measure and k-additive measures, introduced by Sugeno[2] and Grabisch[3] respectively.
20.4.1 Sugeno λ-measure
The Sugeno λ -measure is a special case of fuzzy measures defined iteratively. It has the following definition:
Definition
X Let X = {x1, . . . , xn} be a finite set and let λ ∈ (−1, +∞) .A Sugeno λ -measure is a function g : 2 → [0, 1] such that
1. g(X) = 1 . 2. if A, B ⊆ X (alternatively A, B ∈ 2X ) with A ∩ B = ∅ then g(A ∪ B) = g(A) + g(B) + λg(A)g(B) .
As a convention, the value of g at a singleton set {xi} is called a density and is denoted by gi = g({xi}) . In addition, we have that λ satisfies the property
∏n λ + 1 = (1 + λgi) i=1 Tahani and Keller [4] as well as Wang and Klir have showed that once the densities are known, it is possible to use the previous polynomial to obtain the values of λ uniquely. 20.5. SHAPLEY AND INTERACTION INDICES 63
20.4.2 k-additive fuzzy measure
The k-additive fuzzy measure limits the interaction between the subsets E ⊆ X to size |E| = k . This drastically reduces the number of variables needed to define the fuzzy measure, and as k can be anything from 1 (in which case the fuzzy measure is additive) to X, it allows for a compromise between modelling ability and simplicity.
Definition
A discrete fuzzy measure g on a set X is called k-additive ( 1 ≤ k ≤ |X| ) if its Möbius representation verifies M(E) = 0 , whenever |E| > k for any E ⊆ X , and there exists a subset F with k elements such that M(F ) ≠ 0 .
20.5 Shapley and interaction indices
In game theory, the Shapley value or Shapley index is used to indicate the weight of a game. Shapley values can calculated for fuzzy measures in order to give some indication of the importance of each singleton. In the case of additive fuzzy measures, the Shapley value will be the same as each singleton. For a given fuzzy measure g, and |X| = n , the Shapley index for every i, . . . , n ∈ X is:
∑ (n − |E| − 1)!|E|! ϕ(i) = [g(E ∪ {i}) − g(E)]. n! E⊆X\{i}
The Shapley value is the vector ϕ(g) = (ψ(1), . . . , ψ(n)).
20.6 See also
• Probability theory
• Possibility theory
20.7 References
[1] Gustave Choquet (1953). “Theory of Capacities”. Annales de l'Institut Fourier 5: 131–295.
[2] M. Sugeno (1974). “Theory of fuzzy integrals and its applications. Ph.D. thesis”. Tokyo Institute of Technology, Tokyo, Japan.
[3] M. Grabisch (1997). "k-order additive discrete fuzzy measures and their representation”. Fuzzy Sets and Systems 92 (2): 167–189. doi:10.1016/S0165-0114(97)00168-1.
[4] H. Tahani and J. Keller (1990). “Information Fusion in Computer Vision Using the Fuzzy Integral”. IEEE Transactions on Systems, Man and Cybernetic 20 (3): 733–741. doi:10.1109/21.57289.
• Beliakov, Pradera and Calvo, Aggregation Functions: A Guide for Practitioners, Springer, New York 2007.
• Wang, Zhenyuan, and, George J. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1991.
20.8 External links
• Fuzzy Measure Theory at Fuzzy Image Processing Chapter 21
Fuzzy number
A fuzzy number is an generalization of a regular, real number in the sense that it does not refer to one single value but rather to a connected set of possible values, where each possible value has its own weight between 0 and 1. This weight is called the membership function. A fuzzy number is thus a special case of a convex, normalized fuzzy set of the real line.[1] Just like Fuzzy logic is an extension of Boolean logic (which uses absolute truth and falsehood only, and nothing in between), fuzzy numbers are an extension of real numbers. Calculations with fuzzy numbers allow the incorporation of uncertainty on parameters, properties, geometry, initial conditions, etc.
21.1 See also
• Fuzzy set
• Uncertainty
• Interval arithmetic • Random variable
21.2 References
[1] Michael Hanss, 2005. Applied Fuzzy Arithmetic, An Introduction with Engineering Applications. Springer, ISBN 3-540- 24201-5
21.3 External links
• Fuzzy Logic Tutorial • FuzzyNumbers Package for R: CRAN record
21.4 Applications
• A method for asset valuation that uses fuzzy numbers in investment analysis and real option valuation
64 Chapter 22
Fuzzy pay-off method for real option valuation
The fuzzy pay-off method for real option valuation (FPOM or pay-off method) [1] is a new method for valuing real options, created in 2008. It is based on the use of fuzzy logic and fuzzy numbers for the creation of the possible pay-off distribution of a project (real option). The structure of the method is similar to the probability theory based Datar–Mathews method for real option valuation,[2][3] but the method is not based on probability theory and uses fuzzy numbers and possibility theory in framing the real option valuation problem.
22.1 Method
The Fuzzy pay-off method derives the real option value from a pay-off distribution that is created by using three or four cash-flow scenarios (most often created by an expert or a group of experts). The pay-off distribution is created simply by assigning each of the three cash-flow scenarios a corresponding definition with regards to a fuzzy number (triangular fuzzy number for three scenarios and a trapezoidal fuzzy number for four scenarios). This means that the pay-off distribution is created without any simulation whatsoever. This makes the procedure easy and transparent. The scenarios used are a minimum possible scenario (the lowest possible outcome), the maximum possible scenario (the highest possible outcome) and a best estimate (most likely to happen scenario) that is mapped as a fully possible scenario with a full degree of membership in the set of possible outcomes, or in the case of four scenarios used - two best estimate scenarios that are the upper and lower limit of the interval that is assigned a full degree of membership in the set of possible outcomes. The main observations that lie behind the model for deriving the real option value are the following:
1. The fuzzy NPV of a project is (equal to) the pay-off distribution of a project value that is calculated with fuzzy numbers. 2. The mean value of the positive values of the fuzzy NPV is the “possibilistic” mean value of the positive fuzzy NPV values. 3. Real option value, ROV, calculated from the fuzzy NPV is the “possibilistic” mean value[4] of the positive fuzzy NPV values multiplied with the positive area of the fuzzy NPV over the total area of the fuzzy NPV.
The real option formula can then be written simply as:
A(Pos) ROV = × E[A ] A(Pos) + A(Neg) +
where A(Pos) is the area of the positive part of the fuzzy distribution, A(Neg) is the area of the negative part of the fuzzy distribution, and E[A₊] is the mean value of the positive part of the distribution. It can be seen that when the distribution is totally positive, the real options value reduces to the expected (mean) value, E[A₊].
65 66 CHAPTER 22. FUZZY PAY-OFF METHOD FOR REAL OPTION VALUATION
As can be seen, the real option value can be derived directly from the fuzzy NPV, without simulation.[5] At the same time, simulation is not an absolutely necessary step in the Datar–Mathews method, so the two methods are not very different in that respect. But what is totally different is that the Datar–Mathews method is based on probability theory and as such has a very different foundation from the pay-off method that is based on possibility theory: the way that the two models treat uncertainty is fundamentally different.
22.2 Use of the method
The pay-off method for real option valuation is very easy to use compared to the other real option valuation methods and it can be used with the most commonly used spreadsheet software without any add-ins. The method is useful in analyses for decision making regarding investments that have an uncertain future, and especially so if the underlying data is in the form of cash-flow scenarios. The method is less useful if optimal timing is the objective. The method is flexible and accommodates easily both one-stage investments and multi-stage investments (compound real options). The method has been taken into use in some large international industrial companies for the valuation of research and development projects and portfolios.[6] In these analyses triangular fuzzy numbers are used. Other uses of the method so far are, for example, R&D project valuation IPR valuation, valuation of M&A targets and expected synergies,[7] valuation and optimization of M&A strategies, valuation of area development (construction) projects, valuation of large industrial real investments. The use of the pay-off method is lately taught within the larger framework of real options, for example at the Lappeenranta University of Technology and at the Tampere University of Technology in Finland.
22.3 References
[1] Collan, M., Fullér, R., and Mezei, J., 2009, Fuzzy Pay-Off Method for Real Option Valuation, Journal of Applied Mathe- matics and Decision Sciences, vol. 2009
[2] Datar, V. & Mathews, S. 2004. European Real Options: An Intuitive Algorithm for the Black Scholes Formula. Journal of Applied Finance, 14(1)
[3] Mathews, S. & Datar, V. 2007. A Practical Method for Valuing Real Options: The Boeing Approach. Journal of Applied Corporate Finance, 19(2): 95–104.
[4] Fuller, R. & Majlender, P. 2003. On weighted possibilistic mean and variance of fuzzy numbers. Fuzzy Sets and Systems, 136: 363–374.
[5] Collan, M., Fullér, R., and Mezei, J., 2009, Fuzzy Pay-Off Method for Real Option Valuation, Journal of Applied Mathe- matics and Decision Sciences, vol. 2009
[6] Heikkilä, M., 2009, Selection of R&D Portfolios of Real Options with Fuzzy Pay-offs under Bounded Rationality, IAMSR Research Report, 1/2009, ISBN 978-952-12-2316-7
[7] Kinnunen, J., 2010, Valuing M&A Synergies as (Fuzzy) Real Options, 14th Annual International Conference on Real Options in Rome, Italy, June 16–19, 2010
22.4 External links
• Pay-off Method for ROV Homepage • Powerpoint overview
• A Fuzzy Pay-Off Method for Real Option Valuation, Journal of Applied Mathematics and Decision Sciences (Original Journal Publication) • A Fuzzy Pay-Off Method for Real Option Valuation, IEEE BIFE Conference paper
• Book on the pay-off method, with application examples Chapter 23
Fuzzy routing
Fuzzy routing is the application of fuzzy logic to routing protocols, particularly in the context of ad-hoc wireless networks and in networks supporting multiple quality of service classes. It is currently the subject of research.
23.1 See also
• Dynamic routing • List of ad hoc routing protocols
23.2 External links
• Hui Liu et al., An Adaptive Genetic Fuzzy Multi-path Routing Protocol for Wireless Ad Hoc Networks
• Runtong Zhang, A Fuzzy Routing Mechanism In Next-Generation Networks
67 Chapter 24
Fuzzy rule
A fuzzy rule is defined as a conditional statement in the form:
IF x is A THEN y is B where x and y are linguistic variables; A and B are linguistic values determined by fuzzy sets on the universe of discourse X and Y, respectively.
24.1 Comparison between Boolean and fuzzy logic rules
A classical IF-THEN statement uses binary logic, for instance:
IF man_height is > 180cm THEN man_weight is > 50kg
24.2 Comparison between computational verb and fuzzy logic rules
Computational verb rules(verb rules, for short) are expressed in computational verb logic. The difference between verb and fuzzy rules is that the former using verbs other than BE in the statement while the latter using verb BE only. For example, the two fuzzy rules in above have the following corresponding computational verb counterparts:
• IF man_height becomes tall THEN man_weight become heavy;
• IF man_height increase to tall THEN man_weight probably grow to heavy;
24.3 See also
• Fuzzy logic
• Computational verb logic
68 Chapter 25
Fuzzy set
In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced by Lotfi A. Zadeh[1] and Dieter Klaua[2] in 1965 as an extension of the classical notion of set. At the same time, Salii (1965) defined a more general kind of structures called L-relations, which he studied in an abstract algebraic context. Fuzzy relations, which are used now in different areas, such as linguistics (De Cock, et al., 2000), decision-making (Kuzmin, 1982) and clustering (Bezdek, 1978), are special cases of L-relations when L is the unit interval [0, 1]. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition — an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1]. Fuzzy sets generalize classical sets, since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1.[3] In fuzzy set theory, classical bivalent sets are usually called crisp sets. The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise, such as bioinformatics.[4] It has been suggested by Thayer Watkins that Zadeh’s ethnicity is an example of a fuzzy set because “His father was Turkish-Iranian (Azerbaijani) and his mother was Russian. His father was a journalist working in Baku, Azerbaijan in the Soviet Union...Lotfi was born in Baku in 1921 and lived there until his family moved to Tehran in 1931.”[5]
25.1 Definition
A fuzzy set is a pair (U, m) where U is a set and m: U → [0, 1].
For each x ∈ U, the value m(x) is called the grade of membership of x in (U, m). For a finite set U = {x1, . . . , xn}, the fuzzy set (U, m) is often denoted by {m(x1)/x1, . . . , m(xn)/xn}. Let x ∈ U. Then x is called not included in the fuzzy set (U, m) if m(x) = 0 , x is called fully included if m(x) = 1 , and x is called a fuzzy member if 0 < m(x) < 1 .[6] The set {x ∈ U | m(x) > 0} is called the support of (U, m) and the set {x ∈ U | m(x) = 1} is called its kernel or core. The function m is called the membership function of the fuzzy set (U, m). Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a (fixed or variable) algebra or structure L of a given kind; usually it is required that L be at least a poset or lattice. These are usually called L-fuzzy sets, to distinguish them from those valued over the unit interval. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions. These kinds of generalizations were first considered in 1967 by Joseph Goguen, who was a student of Zadeh.[7]
25.2 Fuzzy logic
Main article: Fuzzy logic
As an extension of the case of multi-valued logic, valuations ( µ : Vo → W ) of propositional variables ( Vo ) into a set of membership degrees ( W ) can be thought of as membership functions mapping predicates into fuzzy sets (or
69 70 CHAPTER 25. FUZZY SET more formally, into an ordered set of fuzzy pairs, called a fuzzy relation). With these valuations, many-valued logic can be extended to allow for fuzzy premises from which graded conclusions may be drawn.[8] This extension is sometimes called “fuzzy logic in the narrow sense” as opposed to “fuzzy logic in the wider sense,” which originated in the engineering fields of automated control and knowledge engineering, and which encompasses many topics involving fuzzy sets and “approximated reasoning.”[9] Industrial applications of fuzzy sets in the context of “fuzzy logic in the wider sense” can be found at fuzzy logic.
25.3 Fuzzy number
Main article: Fuzzy number
A fuzzy number is a convex, normalized fuzzy set A˜ ⊆ R whose membership function is at least segmentally continuous and has the functional value µA(x) = 1 at precisely one element. This can be likened to the funfair game “guess your weight,” where someone guesses the contestant’s weight, with closer guesses being more correct, and where the guesser “wins” if he or she guesses near enough to the contestant’s weight, with the actual weight being completely correct (mapping to 1 by the membership function).
25.4 Fuzzy interval
A fuzzy interval is an uncertain set A˜ ⊆ R with a mean interval whose elements possess the membership function value µA(x) = 1 . As in fuzzy numbers, the membership function must be convex, normalized, at least segmentally continuous.[10]
25.5 Fuzzy relation equation
The fuzzy relation equation is an equation of the form A · R = B, where A and B are fuzzy sets, R is a fuzzy relation, and A · R stands for the composition of A with R.
25.6 Axiomatic definition of credibility
[11] Let A be a non-empty set and P(A) be the power set of A . The set function Cr is known as credibility measure if it satisfies following condition
• Axiom 1: Cr{A} = 1
• Axiom 2: If B is subset of C, then, Cr{B} ≤ Cr{C}
• Axiom 3: Cr{B} + Cr{Bc} = 1
• {∪ } { } Axiom 4: Cr Ai = supi(Cr(Ai)) , for any event Ai with supi Cr Ai < 0.5
Cr{B} indicates how frequently event B will occur.
25.7 Credibility inversion theorem
[12] Let A be a fuzzy variable with membership function u. Then for any set B of real numbers, we have
( ) 1 Cr{A ∈ B} = sup u(t) + 1 − sup u(t) 2 t∈B t∈Bc 25.8. EXPECTED VALUE 71
25.8 Expected Value
[13] Let A be a fuzzy variable. Then the expected value is
∫ ∫ ∞ 0 E[A] = Cr{A ≥ t} dt − Cr{A ≤ t} dt. 0 −∞
25.9 Entropy
[14] Let A be a fuzzy variable with a continuous membership function. Then its entropy is
∫ ∞ H[A] = S(Cr{A ≥ t}) dt. −∞
Where
S(y) = −y lny − (1 − y) ln(1 − y)
25.10 Generalizations
There are many mathematical constructions similar to or more general than fuzzy sets. Since fuzzy sets were intro- duced in 1965, a lot of new mathematical constructions and theories treating imprecision, inexactness, ambiguity, and uncertainty have been developed. Some of these constructions and theories are extensions of fuzzy set theory, while others try to mathematically model imprecision and uncertainty in a different way (Burgin and Chunihin, 1997; Kerre, 2001; Deschrijver and Kerre, 2003). The diversity of such constructions and corresponding theories includes:
• interval sets (Moore, 1966),
• L-fuzzy sets (Goguen, 1967),
• flou sets (Gentilhomme, 1968),
• Boolean-valued fuzzy sets (Brown, 1971),
• type-2 fuzzy sets and type-n fuzzy sets (Zadeh, 1975),
• set-valued sets (Chapin, 1974; 1975),
• interval-valued fuzzy sets (Grattan-Guinness, 1975; Jahn, 1975; Sambuc, 1975; Zadeh, 1975),
• functions as generalizations of fuzzy sets and multisets (Lake, 1976),
• level fuzzy sets (Radecki, 1977)
• underdetermined sets (Narinyani, 1980),
• rough sets (Pawlak, 1982),
• intuitionistic fuzzy sets (Atanassov, 1983),
• fuzzy multisets (Yager, 1986),
• intuitionistic L-fuzzy sets (Atanassov, 1986),
• rough multisets (Grzymala-Busse, 1987), 72 CHAPTER 25. FUZZY SET
• fuzzy rough sets (Nakamura, 1988),
• real-valued fuzzy sets (Blizard, 1989),
• vague sets (Wen-Lung Gau and Buehrer, 1993),
• Q-sets (Gylys, 1994)
• shadowed sets (Pedrycz, 1998),
• α-level sets (Yao, 1997),
• genuine sets (Demirci, 1999),
• soft sets (Molodtsov, 1999),
• intuitionistic fuzzy rough sets (Cornelis, De Cock and Kerre, 2003)
• blurry sets (Smith, 2004)
• L-fuzzy rough sets (Radzikowska and Kerre, 2004),
• generalized rough fuzzy sets (Feng, 2010)
• rough intuitionistic fuzzy sets (Thomas and Nair, 2011),
• soft rough fuzzy sets (Meng, Zhang and Qin, 2011)
• soft fuzzy rough sets (Meng, Zhang and Qin, 2011)
• soft multisets (Alkhazaleh, Salleh and Hassan, 2011)
• fuzzy soft multisets (Alkhazaleh and Salleh, 2012)
25.11 See also
• Alternative set theory
• Defuzzification
• Fuzzy concept
• Fuzzy mathematics
• Fuzzy measure theory
• Fuzzy set operations
• Fuzzy subalgebra
• Linear partial information
• Neuro-fuzzy
• Rough fuzzy hybridization
• Rough set
• Sørensen similarity index
• Type-2 Fuzzy Sets and Systems
• Uncertainty
• Interval finite element
• Multiset 25.12. REFERENCES 73
25.12 References
[1] L. A. Zadeh (1965) “Fuzzy sets”. Information and Control 8 (3) 338–353.
[2] Klaua, D. (1965) Über einen Ansatz zur mehrwertigen Mengenlehre. Monatsb. Deutsch. Akad. Wiss. Berlin 7, 859–876. A recent in-depth analysis of this paper has been provided by Gottwald, S. (2010). “An early approach toward graded iden- tity and graded membership in set theory”. Fuzzy Sets and Systems 161 (18): 2369–2379. doi:10.1016/j.fss.2009.12.005.
[3] D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York.
[4] Lily R. Liang, Shiyong Lu, Xuena Wang, Yi Lu, Vinay Mandal, Dorrelyn Patacsil, and Deepak Kumar, “FM-test: A Fuzzy-Set-Theory-Based Approach to Differential Gene Expression Data Analysis”, BMC Bioinformatics, 7 (Suppl 4): S7. 2006.
[5] “Fuzzy Logic: The Logic of Fuzzy Sets”
[6] AAAI
[7] Goguen, Joseph A., 196, "L-fuzzy sets”. Journal of Mathematical Analysis and Applications 18: 145–174
[8] Siegfried Gottwald, 2001. A Treatise on Many-Valued Logics. Baldock, Hertfordshire, England: Research Studies Press Ltd., ISBN 978-0-86380-262-1
[9] “The concept of a linguistic variable and its application to approximate reasoning,” Information Sciences 8: 199–249, 301–357; 9: 43–80.
[10] “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets and Systems 1: 3–28
[11] Liu, Baoding. “Uncertain theory: an introduction to its axiomatic foundations.” Berlin: Springer-Verlag (2004).
[12] Liu, Baoding, and Yian-Kui Liu. “Expected value of fuzzy variable and fuzzy expected value models.” Fuzzy Systems, IEEE Transactions on 10.4 (2002): 445-450.
[13] Liu, Baoding, and Yian-Kui Liu. “Expected value of fuzzy variable and fuzzy expected value models.” Fuzzy Systems, IEEE Transactions on 10.4 (2002): 445-450.
[14] Xuecheng, Liu. “Entropy, distance measure and similarity measure of fuzzy sets and their relations.” Fuzzy sets and systems 52.3 (1992): 305-318.
25.13 Further reading
• Alkhazaleh, S. and Salleh, A.R. Fuzzy Soft Multiset Theory, Abstract and Applied Analysis, 2012, article ID 350600, 20 p.
• Alkhazaleh, S., Salleh, A.R. and Hassan, N. Soft Multisets Theory, Applied Mathematical Sciences, v. 5, No. 72, 2011, pp. 3561–3573
• Atanassov, K. T. (1983) Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia (deposited in Central Sci.-Technical Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian)
• Atanasov, K. (1986) Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, v. 20, No. 1, pp. 87–96
• Bezdek, J.C. (1978) Fuzzy partitions and relations and axiomatic basis for clustering, Fuzzy Sets and Systems, v.1, pp. 111–127
• Blizard, W.D. (1989) Real-valued Multisets and Fuzzy Sets, Fuzzy Sets and Systems, v. 33, pp. 77–97
• Brown, J.G. (1971) A Note on Fuzzy Sets, Information and Control, v. 18, pp. 32–39
• Chapin, E.W. (1974) Set-valued Set Theory, I, Notre Dame J. Formal Logic, v. 15, pp. 619–634
• Chapin, E.W. (1975) Set-valued Set Theory, II, Notre Dame J. Formal Logic, v. 16, pp. 255–267
• Chris Cornelis, Martine De Cock and Etienne E. Kerre, Intuitionistic fuzzy rough sets: at the crossroads of imperfect knowledge, Expert Systems, v. 20, issue 5, pp. 260–270, 2003 74 CHAPTER 25. FUZZY SET
• Cornelis, C., Deschrijver, C., and Kerre, E. E. (2004) Implication in intuitionistic and interval-valued fuzzy set theory: construction, classification, application, International Journal of Approximate Reasoning, v. 35, pp. 55–95 • Martine De Cock, Ulrich Bodenhofer, and Etienne E. Kerre, Modelling Linguistic Expressions Using Fuzzy Relations, (2000) Proceedings 6th International Conference on Soft Computing. Iizuka 2000, Iizuka, Japan (1–4 October 2000) CDROM. p. 353-360 • Demirci, M. (1999) Genuine Sets, Fuzzy Sets and Systems, v. 105, pp. 377–384 • Deschrijver, G. and Kerre, E.E. On the relationship between some extensions of fuzzy set theory, Fuzzy Sets and Systems, v. 133, no. 2, pp. 227–235, 2003 • Didier Dubois, Henri M. Prade, ed. (2000). Fundamentals of fuzzy sets. The Handbooks of Fuzzy Sets Series 7. Springer. ISBN 978-0-7923-7732-0. • Feng F. Generalized Rough Fuzzy Sets Based on Soft Sets, Soft Computing, July 2010, Volume 14, Issue 9, pp 899–911 • Gentilhomme, Y. (1968) Les ensembles flous en linguistique, Cahiers Linguistique Theoretique Appliqee, 5, pp. 47–63 • Gogen, J.A. (1967) L-fuzzy Sets, Journal Math. Analysis Appl., v. 18, pp. 145–174 • Gottwald, S. (2006). “Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part I: Model-Based and Axiomatic Approaches”. Studia Logica 82 (2): 211–244. doi:10.1007/s11225-006-7197-8.. Gottwald, S. (2006). “Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part II: Category Theoretic Approaches”. Studia Logica 84: 23–50. doi:10.1007/s11225-006-9001-1. preprint.. • Grattan-Guinness, I. (1975) Fuzzy membership mapped onto interval and many-valued quantities. Z. Math. Logik. Grundladen Math. 22, pp. 149–160. • Grzymala-Busse, J. Learning from examples based on rough multisets, in Proceedings of the 2nd International Symposium on Methodologies for Intelligent Systems, Charlotte, NC, USA, 1987, pp. 325–332 • Gylys, R. P. (1994) Quantal sets and sheaves over quantales, Liet. Matem. Rink., v. 34, No. 1, pp. 9–31. • Ulrich Höhle, Stephen Ernest Rodabaugh, ed. (1999). Mathematics of fuzzy sets: logic, topology, and measure theory. The Handbooks of Fuzzy Sets Series 3. Springer. ISBN 978-0-7923-8388-8. • Jahn, K. U. (1975) Intervall-wertige Mengen, Math.Nach. 68, pp. 115–132 • Kerre, E.E. A first view on the alternatives of fuzzy set theory, Computational Intelligence in Theory and Practice (B. Reusch, K-H . Temme, eds) Physica-Verlag, Heidelberg (ISBN 3-7908-1357-5), 2001, pp. 55– 72 • George J. Klir; Bo Yuan (1995). Fuzzy sets and fuzzy logic: theory and applications. Prentice Hall. ISBN 978-0-13-101171-7. • Kuzmin,V.B. Building Group Decisions in Spaces of Strict and Fuzzy Binary Relations, Nauka, Moscow, 1982 (in Russian) • Lake, J. (1976) Sets, fuzzy sets, multisets and functions, J. London Math. Soc., II Ser., v. 12, pp. 323–326 • Meng, D., Zhang, X. and Qin, K. Soft rough fuzzy sets and soft fuzzy rough sets, 'Computers & Mathematics with Applications’, v. 62, issue 12, 2011, pp. 4635–4645 • Miyamoto, S. Fuzzy Multisets and their Generalizations, in 'Multiset Processing', LNCS 2235, pp. 225–235, 2001 • Molodtsov, O. (1999) Soft set theory – first results, Computers & Mathematics with Applications, v. 37, No. 4/5, pp. 19–31 • Moore, R.E. Interval Analysis, New York, Prentice-Hall, 1966 • Nakamura, A. (1988) Fuzzy rough sets, 'Notes on Multiple-valued Logic in Japan', v. 9, pp. 1–8 25.14. EXTERNAL LINKS 75
• Narinyani, A.S. Underdetermined Sets – A new datatype for knowledge representation, Preprint 232, Project VOSTOK, issue 4, Novosibirsk, Computing Center, USSR Academy of Sciences, 1980 • Pedrycz, W. Shadowed sets: representing and processing fuzzy sets, IEEE Transactions on System, Man, and Cybernetics, Part B, 28, 103-109, 1998. • Radecki, T. Level Fuzzy Sets, 'Journal of Cybernetics’, Volume 7, Issue 3-4, 1977
• Radzikowska, A.M. and Etienne E. Kerre, E.E. On L-Fuzzy Rough Sets, Artificial Intelligence and Soft Computing - ICAISC 2004, 7th International Conference, Zakopane, Poland, June 7–11, 2004, Proceedings; 01/2004 • Salii, V.N. (1965) Binary L-relations, Izv. Vysh. Uchebn. Zaved., Matematika, v. 44, No.1, pp. 133–145 (in Russian) • Sambuc, R. Fonctions φ-floues: Application a l'aide au diagnostic en pathologie thyroidienne, Ph. D. Thesis Univ. Marseille, France, 1975.
• Seising, Rudolf: The Fuzzification of Systems. The Genesis of Fuzzy Set Theory and Its Initial Applications— Developments up to the 1970s (Studies in Fuzziness and Soft Computing, Vol. 216) Berlin, New York, [et al.]: Springer 2007. • Smith, N.J.J. (2004) Vagueness and blurry sets, 'J. of Phil. Logic', 33, pp. 165–235
• Thomas, K.V. and L. S. Nair, Rough intuitionistic fuzzy sets in a lattice, 'International Mathematical Forum', Vol. 6, 2011, no. 27, 1327 - 1335
• Yager, R. R. (1986) On the Theory of Bags, International Journal of General Systems, v. 13, pp. 23–37 • Yao, Y.Y., Combination of rough and fuzzy sets based on α-level sets, in: Rough Sets and Data Mining: Analysis for Imprecise Data, Lin, T.Y. and Cercone, N. (Eds.), Kluwer Academic Publishers, Boston, pp. 301–321, 1997.
• Y. Y. Yao, A comparative study of fuzzy sets and rough sets, Information Sciences, v. 109, Issue 1-4, 1998, pp. 227 – 242
• Zadeh, L. (1975) The concept of a linguistic variable and its application to approximate reasoning–I, Inform. Sci., v. 8, pp. 199–249
• Hans-Jürgen Zimmermann (2001). Fuzzy set theory—and its applications (4th ed.). Kluwer. ISBN 978-0- 7923-7435-0.
• Gianpiero Cattaneo and Davide Ciucci, “Heyting Wajsberg Algebras as an Abstract Environment Linking Fuzzy and Rough Sets” in J.J. Alpigini et al. (Eds.): RSCTC 2002, LNAI 2475, pp. 77–84, 2002. doi:10.1007/3- 540-45813-1_10
25.14 External links
• Uncertainty model Fuzziness • Fuzzy Systems Journal
• ScholarPedia • The Algorithm of Fuzzy Analysis
• Fuzzy Image Processing Chapter 26
Fuzzy set operations
A fuzzy set operation is an operation on fuzzy sets. These operations are generalization of crisp set operations. There is more than one possible generalization. The most widely used operations are called standard fuzzy set operations. There are three operations: fuzzy complements, fuzzy intersections, and fuzzy unions.
26.1 Standard fuzzy set operations
Let A and B be fuzzy sets that A,B ∈ U, u is an element in the U universe (e.g. value)
Standard complement
Standard intersection
Standard union
26.2 Fuzzy complements
A(x) is defined as the degree to which x belongs to A. Let cA denote a fuzzy complement of A of type c. Then cA(x) is the degree to which x belongs to cA, and the degree to which x does not belong to A.(A(x) is therefore the degree to which x does not belong to cA.) Let a complement cA be defined by a function
76 26.3. FUZZY INTERSECTIONS 77
c : [0,1] → [0,1]
c(A(x)) = cA(x)
26.2.1 Axioms for fuzzy complements
Axiom c1. Boundary condition c(0) = 1 and c(1) = 0
Axiom c2. Monotonicity For all a, b ∈ [0, 1], if a < b, then c(a) > c(b)
Axiom c3. Continuity c is continuous function.
Axiom c4. Involutions c is an involution, which means that c(c(a)) = a for each a ∈ [0,1]
26.3 Fuzzy intersections
Main article: T-norm
The intersection of two fuzzy sets A and B is specified in general by a binary operation on the unit interval, a function of the form
i:[0,1]×[0,1] → [0,1].
(A ∩ B)(x) = i[A(x), B(x)] for all x.
26.3.1 Axioms for fuzzy intersection
Axiom i1. Boundary condition i(a, 1) = a
Axiom i2. Monotonicity b ≤ d implies i(a, b) ≤ i(a, d)
Axiom i3. Commutativity i(a, b) = i(b, a)
Axiom i4. Associativity i(a, i(b, d)) = i(i(a, b), d)
Axiom i5. Continuity i is a continuous function
Axiom i6. Subidempotency i(a, a) ≤ a
26.4 Fuzzy unions
The union of two fuzzy sets A and B is specified in general by a binary operation on the unit interval function of the form
u:[0,1]×[0,1] → [0,1].
(A ∪ B)(x) = u[A(x), B(x)] for all x 78 CHAPTER 26. FUZZY SET OPERATIONS
26.4.1 Axioms for fuzzy union
Axiom u1. Boundary condition u(a, 0) =u(0 ,a) = a
Axiom u2. Monotonicity b ≤ d implies u(a, b) ≤ u(a, d)
Axiom u3. Commutativity u(a, b) = u(b, a)
Axiom u4. Associativity u(a, u(b, d)) = u(u(a, b), d)
Axiom u5. Continuity u is a continuous function
Axiom u6. Superidempotency u(a, a) ≥ a
Axiom u7. Strict monotonicity a1 < a2 and b1 < b2 implies u(a1, b1) < u(a2, b2)
26.5 Aggregation operations
Aggregation operations on fuzzy sets are operations by which several fuzzy sets are combined in a desirable way to produce a single fuzzy set. Aggregation operation on n fuzzy set (2 ≤ n) is defined by a function
h:[0,1]n → [0,1]
26.5.1 Axioms for aggregation operations fuzzy sets
Axiom h1. Boundary condition h(0, 0, ..., 0) = 0 and h(1, 1, ..., 1) = 1
Axiom h2. Monotonicity For any pair
Axiom h3. Continuity h is a continuous function.
26.6 See also
• Fuzzy logic • Fuzzy set • T-norm • Type-2 fuzzy sets and systems
26.7 Further reading
• Klir, George J.; Bo Yuan (1995). Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall. ISBN 978-0131011717.
26.8 External References
• L.A. Zadeh. Fuzzy sets. Information and Control, 8:338–353, 1965 Chapter 27
Fuzzy Sets and Systems
Fuzzy Sets and Systems is a peer-reviewed international scientific journal published by Elsevier on behalf of the International Fuzzy Systems Association (IFSA) and was founded in 1978. The editors-in-chief (as of 2010) are Bernard De Baets of the Dept. of Applied Mathematics, Biometrics and Process Control, (at the University of Gent in Belgium), Didier Dubois (of IRIT, Université Paul Sabatier in Toulouse, France) and Eyke Hüllermeier (of the Dept. of Mathematics and Computer Science, Universität Marburg, Germany). The journal publishes 24 issues a year. Fuzzy Sets and Systems is abstracted and indexed by Scopus and the Science Citation Index. According to the Journal Citation Reports released in 2010, its 2-year impact factor calculated for 2009 is 2.138 and its 2010 5-year impact factor for 2009 is 2.551.
27.1 See also
• Fuzzy Control System
• Fuzzy Control Language
• Fuzzy logic • Fuzzy set
79 Chapter 28
Fuzzy subalgebra
Fuzzy subalgebras theory is a chapter of fuzzy set theory. It is obtained from an interpretation in a multi-valued logic of axioms usually expressing the notion of subalgebra of a given algebraic structure.
28.1 Definition
Consider a first order language for algebraic structures with a monadic predicate symbol S. Then a fuzzy subalgebra is a fuzzy model of a theory containing, for any n-ary operation h, the axioms
∀x1, ..., ∀xn(S(x1) ∧ ..... ∧ S(xn) → S(h(x1, ..., xn)) and, for any constant c, S(c). The first axiom expresses the closure of S with respect to the operation h, and the second expresses the fact that c is an element in S. As an example, assume that the valuation structure is defined in [0,1] and denote by ⊙ the operation in [0,1] used to interpret the conjunction. Then a fuzzy subalgebra of an algebraic structure whose domain is D is defined by a fuzzy subset s : D → [0,1] of D such that, for every d1,...,d in D, if h is the interpretation of the n-ary operation symbol h, then
• s(d1) ⊙ ... ⊙ s(dn) ≤ s(h(d1, ..., dn))
Moreover, if c is the interpretation of a constant c such that s(c) = 1. A largely studied class of fuzzy subalgebras is the one in which the operation ⊙ coincides with the minimum. In such a case it is immediate to prove the following proposition. Proposition. A fuzzy subset s of an algebraic structure defines a fuzzy subalgebra if and only if for every λ in [0,1], the closed cut {x ∈ D : s(x)≥ λ} of s is a subalgebra.
28.2 Fuzzy subgroups and submonoids
The fuzzy subgroups and the fuzzy submonoids are particularly interesting classes of fuzzy subalgebras. In such a case a fuzzy subset s of a monoid (M,•,u) is a fuzzy submonoid if and only if
1. s(u) = 1
2. s(x) ⊙ s(y) ≤ s(x · y)
where u is the neutral element in A. Given a group G, a fuzzy subgroup of G is a fuzzy submonoid s of G such that
• s(x) ≤ s(x−1).
80 28.3. BIBLIOGRAPHY 81
It is possible to prove that the notion of fuzzy subgroup is strictly related with the notions of fuzzy equivalence. In fact, assume that S is a set, G a group of transformations in S and (G,s) a fuzzy subgroup of G. Then, by setting
• e(x,y) = Sup{s(h) : h is an element in G such that h(x) = y} we obtain a fuzzy equivalence. Conversely, let e be a fuzzy equivalence in S and, for every transformation h of S, set
• s(h)= Inf{e(x,h(x)): x∈S}.
Then s defines a fuzzy subgroup of transformation in S. In a similar way we can relate the fuzzy submonoids with the fuzzy orders.
28.3 Bibliography
• Klir, G. and Bo Yuan, Fuzzy Sets and Fuzzy Logic (1995) ISBN 978-0-13-101171-7 • Zimmermann H., Fuzzy Set Theory and its Applications (2001), ISBN 978-0-7923-7435-0.
• Chakraborty H. and Das S., On fuzzy equivalence 1, Fuzzy Sets and Systems, 11 (1983), 185-193. • Demirci M., Recasens J., Fuzzy groups, fuzzy functions and fuzzy equivalence relations, Fuzzy Sets and Sys- tems, 144 (2004), 441-458. • Di Nola A., Gerla G., Lattice valued algebras, Stochastica, 11 (1987), 137-150.
• Hájek P., Metamathematics of fuzzy logic. Kluwer 1998. • Klir G., UTE H. St.Clair and Bo Yuan Fuzzy Set Theory Foundations and Applications,1997.
• Gerla G., Scarpati M., Similarities, Fuzzy Groups: a Galois Connection, J. Math. Anal. Appl., 292 (2004), 33-48.
• Mordeson J., Kiran R. Bhutani and Azriel Rosenfeld. Fuzzy Group Theory, Springer Series: Studies in Fuzzi- ness and Soft Computing, Vol. 182, 2005.
• Rosenfeld A., Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512-517. • Zadeh L.A., Fuzzy Sets, ‘’Information and Control’’, 8 (1965) 338353.
• Zadeh L.A., Similarity relations and fuzzy ordering, Inform. Sci. 3 (1971) 177–200. Chapter 29
Fuzzy transportation
The aim of fuzzy transportation is to find the least transportation cost of some commodities through a capacitated network when the supply and demand of nodes and the capacity and cost of edges are represented as fuzzy numbers. This problem is a new branch in combinatorial optimization and network flow problems. Combinatorial algorithms can be provided to solve fuzzy transportation problem to find the fuzzy optimal flow(s). Such methods are capable of handling the decision maker’s risk taking. Some application of such standpoint were presented in industries. Liu and Kao pursued this attempt to find better solution for this problem (Network flow problems with fuzzy arc lengths, IEEE Transactions on Systems, Man and Cybernetics Part B: Cybernetics, 34 (2004) 765-769). It is interesting to check that which methods in traditional fuzzy optimization problem can be extended to combina- torial optimization problems e.g., transformation that they maintain the nice structure of problem. Then, valuable algorithms can be proposed for fuzzy combinatorial optimization to take the uncertainty of real problems into account. By using fuzzy transportation, it is a reasonable attempt to find special solutions for hazardous material transportation because of the possibility of implementing the optimistic and pessimistic concepts into account.
82 Chapter 30
High Performance Fuzzy Computing
The term High Performance Fuzzy Computing (HPFC) refers to those technologies able to exploit supercomputers and computer clusters to perform high performance Fuzzy Logic computations. Thus HPFC is just a special case of the much more general HPC. In the specific case of fuzzy logic, however, there exist more traditional ways to achieve high performance, that could be considered HPFC but in a broader sense, like the hardware implementations on DSP or FPGA. More recently, another alternative has emerged: fuzzy computing on GPU.
30.1 External links
• (English) Rapid prototyping of high performance fuzzy computing applications using high level GPU program- ming for maritime operations support • (English) Speedup of Implementing Fuzzy Neural Networks with high-dimensional inputs through Parallel Processing on Graphic Processing Units
• (English) Speedup of Fuzzy Clustering Through Stream Processing on Graphics Processing Units • (English) GPUcomputing.net
83 Chapter 31
Linear partial information
Linear partial information (LPI) is a method of making decisions based on insufficient or fuzzy information. LPI was introduced in 1970 by Polish - Swiss mathematician Edward Kofler (1911–2007) to simplify decision processes. Comparing to other methods the LPI-fuzziness is algorithmically simple and particularly in decision making, more practically oriented. Instead of an indicator function the decision maker linearizes any fuzziness by establishing of linear restrictions for fuzzy probability distributions or normalized weights. In the LPI-procedure the decision maker linearizes any fuzziness instead of applying a membership function. This can be done by establishing stochastic and non-stochastic LPI-relations. A mixed stochastic and non-stochastic fuzzification is often a basis for the LPI- procedure. By using the LPI-methods any fuzziness in any decision situation can be considered on the base of the linear fuzzy logic.
31.1 Definition
Any Stochastic Partial Information SPI(p), which can be considered as a solution of a linear inequality system, is called Linear Partial Information LPI(p) about probability p. It can be considered as an LPI-fuzzification of the probability p corresponding to the concepts of linear fuzzy logic.
31.2 Applications
The MaxEmin Principle To obtain the maximally warranted expected value, the decision maker has to choose the strategy which maximizes the minimal expected value. This procedure leads to the MaxEmin - Principle and is an extension of the Bernoulli’s principle.
The MaxWmin Principle This principle leads to the maximal guaranteed weight function, regarding the extreme weights.
The Prognostic Decision Principle (PDP) This principle is based on the prognosis interpretation of strategies un- der fuzziness.
31.3 Fuzzy equilibrium and stability
Despite the fuzziness of information, it is often necessary to choose the optimal, most cautious strategy, for example in economic planning, in conflict situations or in daily decisions. This is impossible without the concept of fuzzy equilibrium. The concept of fuzzy stability is considered as an extension into a time interval, taking into account the corresponding stability area of the decision maker. The more complex is the model, the softer a choice has to be considered. The idea of fuzzy equilibrium is based on the optimization principles. Therefore the MaxEmin-, MaxGmin- and PDP-stability have to be analyzed. The violation of these principles leads often to wrong predictions and decisions.
84 31.4. LPI EQUILIBRIUM POINT 85
31.4 LPI equilibrium point
Considering a given LPI-decision model, as a convolution of the corresponding fuzzy states or a disturbance set, the fuzzy equilibrium strategy remains the most cautious one, despite of the presence of the fuzziness. Any deviation from this strategy can cause a loss for the decision maker.
31.5 See also
• Edward Kofler
• Fuzzy set
• Game theory
• Defuzzification
• Fuzziness
• Stochastic process
• Deterministic
• Probability distribution
• Uncertainty
• Vagueness
• Optimization (mathematics)
• Logic
• List of set theory topics
31.6 Selected references
• Edward Kofler - Equilibrium Points, Stability and Regulation in Fuzzy Optimisation Systems under Linear Partial Stochastic Information (LPI), Proceedings of the International Congress of Cybernetics and Systems, AFCET, Paris 1984, pp. 233–240
• Edward Kofler - Decision Making under Linear Partial Information. Proceedings of the European Congress EUFIT, Aachen, 1994, p. 891-896.
• Edward Kofler - Linear Partial Information with Applications. Proceedings of ISFL 1997 (International Symposium on Fuzzy Logic), Zurich, 1997, p. 235-239.
• Edward Kofler – Entscheidungen bei teilweise bekannter Verteilung der Zustände, Zeitschrift für OR, Vol. 18/3, 1974
• Edward Kofler - Extensive Spiele bei unvollständiger Information, in Information in der Wirtschaft, Gesellschaft für Wirtschafts- und Sozialwissenschaften, Band 126, Berlin 1982
31.7 External links
• Tools for establishing dominance with linear partial information and attribute hierarchy
• Linear Partial Information with applications
• Linear Partial Information (LPI) with applications to the U.S. economic policy 86 CHAPTER 31. LINEAR PARTIAL INFORMATION
• Practical decision making with Linear Partial Information (LPI)
• Stochastic programming with fuzzy linear partial information on probability distribution • One-shot decisions under Linear Partial Information Chapter 32
Membership function (mathematics)
The membership function of a fuzzy set is a generalization of the indicator function in classical sets. In fuzzy logic, it represents the degree of truth as an extension of valuation. Degrees of truth are often confused with probabilities, although they are conceptually distinct, because fuzzy truth represents membership in vaguely defined sets, not like- lihood of some event or condition. Membership functions were introduced by Zadeh in the first paper on fuzzy sets (1965). Zadeh, in his theory of fuzzy sets, proposed using a membership function (with a range covering the interval (0,1)) operating on the domain of all possible values.
32.1 Definition
For any set X , a membership function on X is any function from X to the real unit interval [0, 1] . Membership functions on X represent fuzzy subsets of X . The membership function which represents a fuzzy set A˜ is usually denoted by µA. For an element x of X , the value µA(x) is called the membership degree of x in the fuzzy set A.˜ The membership degree µA(x) quantifies the grade of membership of the element x to the fuzzy set A.˜ The value 0 means that x is not a member of the fuzzy set; the value 1 means that x is fully a member of the fuzzy set. The values between 0 and 1 characterize fuzzy members, which belong to the fuzzy set only partially.
μ( x ) classical (crisp) set A 1.0 fuzzy set Ã
membership function μ(x )
0.0 x
Membership function of a fuzzy set
Sometimes,[1] a more general definition is used, where membership functions take values in an arbitrary fixed algebra or structure L ; usually it is required that L be at least a poset or lattice. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions.
87 88 CHAPTER 32. MEMBERSHIP FUNCTION (MATHEMATICS)
32.2 Capacity
See the article on Capacity of a set for a closely related definition in mathematics. One application of membership functions is as capacities in decision theory. In decision theory, a capacity is defined as a function, ν from S, the set of subsets of some set, into [0, 1] , such that ν is set-wise monotone and is normalized (i.e. ν(∅) = 0, ν(Ω) = 1). This is a generalization of the notion of a probability measure, where the probability axiom of countable additivity is weakened. A capacity is used as a subjective measure of the likelihood of an event, and the "expected value" of an outcome given a certain capacity can be found by taking the Choquet integral over the capacity.
32.3 See also
• Defuzzification
• Fuzzy measure theory • Fuzzy set operations
• Rough set
32.4 References
[1] First in Goguen (1967).
32.5 Bibliography
• Zadeh L.A., 1965, “Fuzzy sets”. Information and Control 8: 338–353.
• Goguen J.A, 1967, "L-fuzzy sets”. Journal of Mathematical Analysis and Applications 18: 145–174
32.6 External links
• Fuzzy Image Processing Chapter 33
Monoidal t-norm logic
Monoidal t-norm based logic (or shortly MTL), the logic of left-continuous t-norms, is one of t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices;[1] it extends the logic of com- mutative bounded integral residuated lattices (known as Höhle’s monoidal logic, Ono’s FLₑ , or intuitionistic logic without contraction) by the axiom of prelinearity.
33.1 Motivation
T-norms are binary functions on the real unit interval [0, 1] which are often used to represent a conjunction connective in fuzzy logic. Every left-continuous t-norm ∗ has a unique residuum, that is, a function ⇒ such that for all x, y, and z,
x ∗ y ≤ z if and only if x ≤ (y ⇒ z).
The residuum of a left-continuous t-norm can explicitly be defined as
(x ⇒ y) = sup{z | z ∗ x ≤ y}. This ensures that the residuum is the largest function such that for all x and y,
x ∗ (x ⇒ y) ≤ y. The latter can be interpreted as a fuzzy version of the modus ponens rule of inference. The residuum of a left- continuous t-norm thus can be characterized as the weakest function that makes the fuzzy modus ponens valid, which makes it a suitable truth function for implication in fuzzy logic. Left-continuity of the t-norm is the necessary and sufficient condition for this relationship between a t-norm conjunction and its residual implication to hold. Truth functions of further propositional connectives can be defined by means of the t-norm and its residuum, for instance the residual negation ¬x = (x ⇒ 0). In this way, the left-continuous t-norm, its residuum, and the truth functions of additional propositional connectives (see the section Standard semantics below) determine the truth values of complex propositional formulae in [0, 1]. Formulae that always evaluate to 1 are then called tautologies with respect to the given left-continuous t-norm ∗, or ∗- tautologies. The set of all ∗- tautologies is called the logic of the t-norm ∗, since these formulae represent the laws of fuzzy logic (determined by the t-norm) which hold (to degree 1) regardless of the truth degrees of atomic formulae. Some formulae are tautologies with respect to all left- continuous t-norms: they represent general laws of propositional fuzzy logic which are independent of the choice of a particular left-continuous t-norm. These formulae form the logic MTL, which can thus be characterized as the logic of left-continuous t-norms.[2]
33.2 Syntax
89 90 CHAPTER 33. MONOIDAL T-NORM LOGIC
33.2.1 Language
The language of the propositional logic MTL consists of countably many propositional variables and the following primitive logical connectives:
• Implication → (binary)
• Strong conjunction ⊗ (binary). The sign & is a more traditional notation for strong conjunction in the literature on fuzzy logic, while the notation ⊗ follows the tradition of substructural logics.
• Weak conjunction ∧ (binary), also called lattice conjunction (as it is always realized by the lattice operation of meet in algebraic semantics). Unlike BL and stronger fuzzy logics, weak conjunction is not definable in MTL and has to be included among primitive connectives.
• Bottom ⊥ (nullary — a propositional constant); 0 or 0 are common alternative signs and zero a common alternative name for the propositional constant (as the constants bottom and zero of substructural logics coincide in MTL).
The following are the most common defined logical connectives:
• Negation ¬ (unary), defined as
¬A ≡ A → ⊥
• Equivalence ↔ (binary), defined as
A ↔ B ≡ (A → B) ∧ (B → A)
In MTL, the definition is equivalent to (A → B) ⊗ (B → A).
• (Weak) disjunction ∨ (binary), also called lattice disjunction (as it is always realized by the lattice operation of join in algebraic semantics), defined as
A ∨ B ≡ ((A → B) → B) ∧ ((B → A) → A)
• Top ⊤ (nullary), also called one and denoted by 1 or 1 (as the constants top and zero of substructural logics coincide in MTL), defined as
⊤ ≡ ⊥ → ⊥
Well-formed formulae of MTL are defined as usual in propositional logics. In order to save parentheses, it is common to use the following order of precedence:
• Unary connectives (bind most closely)
• Binary connectives other than implication and equivalence
• Implication and equivalence (bind most loosely) 33.3. SEMANTICS 91
33.2.2 Axioms
A Hilbert-style deduction system for MTL has been introduced by Esteva and Godo (2001). Its single derivation rule is modus ponens:
from A and A → B derive B.
The following are its axiom schemata:
(MTL1):(A → B) → ((B → C) → (A → C)) (MTL2): A ⊗ B → A (MTL3): A ⊗ B → B ⊗ A (MTL4a): A ∧ B → A (MTL4b): A ∧ B → B ∧ A (MTL4c): A ⊗ (A → B) → A ∧ B (MTL5a):(A → (B → C)) → (A ⊗ B → C) (MTL5b):(A ⊗ B → C) → (A → (B → C)) (MTL6): ((A → B) → C) → (((B → A) → C) → C) (MTL7): ⊥ → A The traditional numbering of axioms, given in the left column, is derived from the numbering of axioms of Há- jek’s basic fuzzy logic BL.[3] The axioms (MTL4a)–(MTL4c) replace the axiom of divisibility (BL4) of BL. The axioms (MTL5a) and (MTL5b) express the law of residuation and the axiom (MTL6) corresponds to the condition of prelinearity. The axioms (MTL2) and (MTL3) of the original axiomatic system were shown to be redundant (Chvalovský, 2012) and (Cintula, 2005). All the other axioms were shown to be independent (Chvalovský, 2012).
33.3 Semantics
Like in other propositional t-norm fuzzy logics, algebraic semantics is predominantly used for MTL, with three main classes of algebras with respect to which the logic is complete:
• General semantics, formed of all MTL-algebras — that is, all algebras for which the logic is sound • Linear semantics, formed of all linear MTL-algebras — that is, all MTL-algebras whose lattice order is linear • Standard semantics, formed of all standard MTL-algebras — that is, all MTL-algebras whose lattice reduct is the real unit interval [0, 1] with the usual order; they are uniquely determined by the function that interprets strong conjunction, which can be any left-continuous t-norm
33.3.1 General semantics
MTL-algebras
Algebras for which the logic MTL is sound are called MTL-algebras. They can be characterized as prelinear commuta- tive bounded integral residuated lattices. In more detail, an algebraic structure (L, ∧, ∨, ∗, ⇒, 0, 1) is an MTL-algebra if
• (L, ∧, ∨, 0, 1) is a bounded lattice with the top element 0 and bottom element 1 • (L, ∗, 1) is a commutative monoid • ∗ and ⇒ form an adjoint pair, that is, z ∗ x ≤ y if and only if z ≤ x ⇒ y, where ≤ is the lattice order of (L, ∧, ∨), for all x, y, and z in L , (the residuation condition) • (x ⇒ y) ∨ (y ⇒ x) = 1 holds for all x and y in L (the prelinearity condition)
Important examples of MTL algebras are standard MTL-algebras on the real unit interval [0, 1]. Further examples include all Boolean algebras, all linear Heyting algebras (both with ∗ = ∧ ), all MV-algebras, all BL-algebras, etc. Since the residuation condition can equivalently be expressed by identities,[4] MTL-algebras form a variety. 92 CHAPTER 33. MONOIDAL T-NORM LOGIC
Interpretation of the logic MTL in MTL-algebras
The connectives of MTL are interpreted in MTL-algebras as follows:
• Strong conjunction by the monoidal operation ∗ • Implication by the operation ⇒ (which is called the residuum of ∗ ) • Weak conjunction and weak disjunction by the lattice operations ∧ and ∨, respectively (usually denoted by the same symbols as the connectives, if no confusion can arise) • The truth constants zero (top) and one (bottom) by the constants 0 and 1 • The equivalence connective is interpreted by the operation ⇔ defined as
x ⇔ y ≡ (x ⇒ y) ∧ (y ⇒ x) Due to the prelinearity condition, this definition is equivalent to one that uses ∗ instead of ∧, thus x ⇔ y ≡ (x ⇒ y) ∗ (y ⇒ x)
• Negation is interpreted by the definable operation −x ≡ x ⇒ 0
With this interpretation of connectives, any evaluation eᵥ of propositional variables in L uniquely extends to an eval- uation e of all well-formed formulae of MTL, by the following inductive definition (which generalizes Tarski’s truth conditions), for any formulae A, B, and any propositional variable p:
e(p) = ev(p) e(⊥) = 0 e(⊤) = 1 e(A ⊗ B) = e(A) ∗ e(B) e(A → B) = e(A) ⇒ e(B) e(A ∧ B) = e(A) ∧ e(B) e(A ∨ B) = e(A) ∨ e(B) e(A ↔ B) = e(A) ⇔ e(B) e(¬A) = e(A) ⇒ 0 Informally, the truth value 1 represents full truth and the truth value 0 represents full falsity; intermediate truth values represent intermediate degrees of truth. Thus a formula is considered fully true under an evaluation e if e(A) = 1. A formula A is said to be valid in an MTL-algebra L if it is fully true under all evaluations in L, that is, if e(A) = 1 for all evaluations e in L. Some formulae (for instance, p → p) are valid in any MTL-algebra; these are called tautologies of MTL. The notion of global entailment (or: global consequence) is defined for MTL as follows: a set of formulae Γ entails a formula A (or: A is a global consequence of Γ), in symbols Γ |= A, if for any evaluation e in any MTL-algebra, whenever e(B) = 1 for all formulae B in Γ, then also e(A) = 1. Informally, the global consequence relation represents the transmission of full truth in any MTL-algebra of truth values.
General soundness and completeness theorems
The logic MTL is sound and complete with respect to the class of all MTL-algebras (Esteva & Godo, 2001):
A formula is provable in MTL if and only if it is valid in all MTL-algebras.
The notion of MTL-algebra is in fact so defined that MTL-algebras form the class of all algebras for which the logic MTL is sound. Furthermore, the strong completeness theorem holds:[5]
A formula A is a global consequence in MTL of a set of formulae Γ if and only if A is derivable from Γ in MTL. 33.4. BIBLIOGRAPHY 93
33.3.2 Linear semantics
Like algebras for other fuzzy logics,[6] MTL-algebras enjoy the following linear subdirect decomposition property:
Every MTL-algebra is a subdirect product of linearly ordered MTL-algebras.
(A subdirect product is a subalgebra of the direct product such that all projection maps are surjective. An MTL-algebra is linearly ordered if its lattice order is linear.) In consequence of the linear subdirect decomposition property of all MTL-algebras, the completeness theorem with respect to linear MTL-algebras (Esteva & Godo, 2001) holds:
• A formula is provable in MTL if and only if it is valid in all linear MTL-algebras.
• A formula A is derivable in MTL from a set of formulae Γ if and only if A is a global consequence in all linear MTL-algebras of Γ.
33.3.3 Standard semantics
Standard are called those MTL-algebras whose lattice reduct is the real unit interval [0, 1]. They are uniquely determined by the real-valued function that interprets strong conjunction, which can be any left-continuous t-norm ∗ . The standard MTL-algebra determined by a left-continuous t-norm ∗ is usually denoted by [0, 1]∗. In [0, 1]∗, implication is represented by the residuum of ∗, weak conjunction and disjunction respectively by the minimum and maximum, and the truth constants zero and one respectively by the real numbers 0 and 1. The logic MTL is complete with respect to standard MTL-algebras; this fact is expressed by the standard completeness theorem (Jenei & Montagna, 2002):
A formula is provable in MTL if and only if it is valid in all standard MTL-algebras.
Since MTL is complete with respect to standard MTL-algebras, which are determined by left-continuous t-norms, MTL is often referred to as the logic of left-continuous t-norms (similarly as BL is the logic of continuous t-norms).
33.4 Bibliography
• Hájek P., 1998, Metamathematics of Fuzzy Logic. Dordrecht: Kluwer.
• Esteva F. & Godo L., 2001, “Monoidal t-norm based logic: Towards a logic of left-continuous t-norms”. Fuzzy Sets and Systems 124: 271–288.
• Jenei S. & Montagna F., 2002, “A proof of standard completeness of Esteva and Godo’s monoidal logic MTL”. Studia Logica 70: 184–192.
• Ono, H., 2003, “Substructural logics and residuated lattices — an introduction”. In F.V. Hendricks, J. Mali- nowski (eds.): Trends in Logic: 50 Years of Studia Logica, Trends in Logic 20: 177–212.
• Cintula P., 2005, “Short note: On the redundancy of axiom (A3) in BL and MTL”. Soft Computing 9: 942.
• Cintula P., 2006, “Weakly implicative (fuzzy) logics I: Basic properties”. Archive for Mathematical Logic 45: 673–704.
• Chvalovský K., 2012, "On the Independence of Axioms in BL and MTL". Fuzzy Sets and Systems 197: 123– 129, doi:10.1016/j.fss.2011.10.018. 94 CHAPTER 33. MONOIDAL T-NORM LOGIC
33.5 References
[1] Ono (2003).
[2] Conjectured by Esteva and Godo who introduced the logic (2001), proved by Jenei and Montagna (2002).
[3] Hájek (1998), Definition 2.2.4.
[4] The proof of Lemma 2.3.10 in Hájek (1998) for BL-algebras can easily be adapted to work for MTL-algebras, too.
[5] A general proof of the strong completeness with respect to all L-algebras for any weakly implicative logic L (which includes MTL) can be found in Cintula (2006).
[6] Cintula (2006). Chapter 34
MV-algebra
In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation ⊕ , a unary operation ¬ , and the constant 0 , satisfying certain axioms. MV-algebras are the algebraic semantics of Łukasiewicz logic; the letters MV refer to the many-valued logic of Łukasiewicz. MV-algebras coincide with the class of bounded commutative BCK algebras.
34.1 Definitions
An MV-algebra is an algebraic structure ⟨A, ⊕, ¬, 0⟩, consisting of
• a non-empty set A, • a binary operation ⊕ on A, • a unary operation ¬ on A, and • a constant 0 denoting a fixed element of A, which satisfies the following identities:
• (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z), • x ⊕ 0 = x, • x ⊕ y = y ⊕ x, • ¬¬x = x, • x ⊕ ¬0 = ¬0, and • ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x.
By virtue of the first three axioms, ⟨A, ⊕, 0⟩ is a commutative monoid. Being defined by identities, MV-algebras form a variety of algebras. The variety of MV-algebras is a subvariety of the variety of BL-algebras and contains all Boolean algebras. An MV-algebra can equivalently be defined (Hájek 1998) as a prelinear commutative bounded integral residuated lattice ⟨L, ∧, ∨, ⊗, →, 0, 1⟩ satisfying the additional identity x ∨ y = (x → y) → y.
34.2 Examples of MV-algebras
A simple numerical example is A = [0, 1], with operations x ⊕ y = min(x + y, 1) and ¬x = 1 − x. In mathematical fuzzy logic, this MV-algebra is called the standard MV-algebra, as it forms the standard real-valued semantics of Łukasiewicz logic.
95 96 CHAPTER 34. MV-ALGEBRA
The trivial MV-algebra has the only element 0 and the operations defined in the only possible way, 0 ⊕ 0 = 0 and ¬0 = 0. The two-element MV-algebra is actually the two-element Boolean algebra {0, 1}, with ⊕ coinciding with Boolean disjunction and ¬ with Boolean negation. In fact adding the axiom x ⊕ x = x to the axioms defining an MV-algebra results in an axiomantization of Boolean algebras.
If instead the axiom added is x ⊕ x ⊕ x = x ⊕ x , then the axioms define the MV3 algebra corresponding to the three-valued Łukasiewicz logic Ł3. Other finite linearly ordered MV-algebras are obtained by restricting the universe and operations of the standard MV-algebra to the set of n equidistant real numbers between 0 and 1 (both included), that is, the set {0, 1/(n − 1), 2/(n − 1),..., 1}, which is closed under the operations ⊕ and ¬ of the standard MV-algebra; these algebras are usually denoted MV . Another important example is Chang’s MV-algebra, consisting just of infinitesimals (with the order type ω) and their co-infinitesimals. Chang also constructed an MV-algebra from an arbitrary totally ordered abelian group G by fixing a positive element u and defining the segment [0, u] as { x ∈ G | 0 ≤ x ≤ u }, which becomes an MV-algebra with x ⊕ y = min(u, x+y) and ¬x = u−x. Furthermore, Chang showed that every linearly ordered MV-algebra is isomorphic to an MV-algebra constructed from a group in this way. D. Mundici extended the above construction to abelian lattice-ordered groups. If G is such a group with strong (order) unit u, then the “unit interval” { x ∈ G | 0 ≤ x ≤ u } can be equipped with ¬x = u−x, x ⊕ y = u∧G (x+y), x ⊗ y = 0∨G(x+y−u). This construction establishes a categorical equivalence between lattice-ordered abelian groups with strong unit and MV-algebras.
34.3 Relation to Łukasiewicz logic
C. C. Chang devised MV-algebras to study many-valued logics, introduced by Jan Łukasiewicz in 1920. In particular, MV-algebras form the algebraic semantics of Łukasiewicz logic, as described below. Given an MV-algebra A, an A-valuation is a homomorphism from the algebra of propositional formulas (in the language consisting of ⊕, ¬, and 0) into A. Formulas mapped to 1 (or ¬ 0) for all A-valuations are called A-tautologies. If the standard MV-algebra over [0,1] is employed, the set of all [0,1]-tautologies determines so-called infinite-valued Łukasiewicz logic. Chang’s (1958, 1959) completeness theorem states that any MV-algebra equation holding in the standard MV-algebra over the interval [0,1] will hold in every MV-algebra. Algebraically, this means that the standard MV-algebra gener- ates the variety of all MV-algebras. Equivalently, Chang’s completeness theorem says that MV-algebras characterize infinite-valued Łukasiewicz logic, defined as the set of [0,1]-tautologies. The way the [0,1] MV-algebra characterizes all possible MV-algebras parallels the well-known fact that identities holding in the two-element Boolean algebra hold in all possible Boolean algebras. Moreover, MV-algebras charac- terize infinite-valued Łukasiewicz logic in a manner analogous to the way that Boolean algebras characterize classical bivalent logic (see Lindenbaum-Tarski algebra). In 1984, Font, Rodriguez and Torrens introduced the Wajsberg algebra as an alternative model for the infinite-valued Łukasiewicz logic. Wajsberg algebras and MV-algebras are isomorphic.[1]
34.3.1 MVn-algebras
In the 1940s Grigore Moisil introduced his Łukasiewicz–Moisil algebras (LMn-algebras) in the hope of giving algebraic semantics for the (finitely) n-valued Łukasiewicz logic. However, in 1956 Alan Rose discovered that for n ≥ 5, the Łukasiewicz–Moisil algebra does not model the Łukasiewicz n-valued logic. Although C. C. Chang published his MV-algebra in 1958, it is faithful model only for the ℵ0-valued (infinitely-many-valued) Łukasiewicz–Tarski logic. For the axiomatically more complicated (finitely) n-valued Łukasiewicz logics, suitable algebras were published in 1977 by Revaz Grigolia and called MVn-algebras.[2] MVn-algebras are a subclass of LMn-algebras; the inclusion is strict for n ≥ 5.[3] The MVn-algebras are MV-algebras which satisfy some additional axioms, just like the n-valued Łukasiewicz logics have additional axioms added to the ℵ0-valued logic. 34.4. RELATION TO FUNCTIONAL ANALYSIS 97
In 1982 Roberto Cignoli published some additional constraints that added to LMn-algebras are proper models for n-valued Łukasiewicz logic; Cignoli called his discovery proper n-valued Łukasiewicz algebras.[4] The LMn-algebras that are also MVn-algebras are precisely Cignoli’s proper n-valued Łukasiewicz algebras.[5]
34.4 Relation to functional analysis
MV-algebras were related by Daniele Mundici to approximately finite-dimensional C*-algebras by establishing a bijective correspondence between all isomorphism classes of AF C*-algebras with lattice-ordered dimension group and all isomorphism classes of countable MV algebras. Some instances of this correspondence include:
34.5 In software
Further information: Multi-adjoint logic programming
There are multiple frameworks implementing fuzzy logic (type II), and most of them implement what has been called a multi-adjoint logic. This is no more than the implementation of a MV-algebra.
34.6 References
[1] http://journal.univagora.ro/download/pdf/28.pdf citing J. M. Font, A. J. Rodriguez, A. Torrens, Wajsberg Algebras, Stochas- tica, VIII, 1, 5-31, 1984
[2] Lavinia Corina Ciungu (2013). Non-commutative Multiple-Valued Logic Algebras. Springer. pp. vii–viii. ISBN 978-3- 319-01589-7.
[3] Iorgulescu, A.: Connections between MVn-algebras and n-valued Łukasiewicz–Moisil algebras—I. Discrete Math. 181, 155–177 (1998) doi:10.1016/S0012-365X(97)00052-6
[4] R. Cignoli, Proper n-Valued Łukasiewicz Algebras as S-Algebras of Łukasiewicz n-Valued Propositional Calculi, Studia Logica, 41, 1982, 3-16, doi:10.1007/BF00373490
[5] http://journal.univagora.ro/download/pdf/28.pdf
• Chang, C. C. (1958) “Algebraic analysis of many-valued logics,” Transactions of the American Mathematical Society 88: 476–490.
• ------(1959) “A new proof of the completeness of the Lukasiewicz axioms,” Transactions of the American Mathematical Society 88: 74–80.
• Cignoli, R. L. O., D'Ottaviano, I. M. L., Mundici, D. (2000) Algebraic Foundations of Many-valued Reasoning. Kluwer.
• Di Nola A., Lettieri A. (1993) “Equational characterization of all varieties of MV-algebras,” Journal of Algebra 221: 123–131.
• Hájek, Petr (1998) Metamathematics of Fuzzy Logic. Kluwer.
• Mundici, D.: Interpretation of AF C*-algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65, 15–63 (1986) doi:10.1016/0022-1236(86)90015-7
34.7 Further reading
• Daniele Mundici, MV-ALGEBRAS. A short tutorial
• D. Mundici (2011). Advanced Łukasiewicz calculus and MV-algebras. Springer. ISBN 978-94-007-0839-6. 98 CHAPTER 34. MV-ALGEBRA
• Mundici, D. The C*-Algebras of Three-Valued Logic. Logic Colloquium ’88, Proceedings of the Colloquium held in Padova 61–77 (1989). doi:10.1016/s0049-237x(08)70262-3 • Cabrer, L. M. & Mundici, D. A Stone-Weierstrass theorem for MV-algebras and unital ℓ-groups. Journal of Logic and Computation (2014). doi:10.1093/logcom/exu023 • Olivia Caramello, Anna Carla Russo (2014) The Morita-equivalence between MV-algebras and abelian ℓ- groups with strong unit
34.8 External links
• Stanford Encyclopedia of Philosophy:"Many-valued logic"—by Siegfried Gottwald. Chapter 35
Noise-based logic
Noise-based logic (NBL)[1][2][3][4][5][6][7][8][8] is a new class of multivalued deterministic logic schemes where the logic values and bits are represented by different realizations of a stochastic process. The concept of noise-based logic and its name was created by Laszlo B. Kish. In its foundation paper[3] it is noted that the idea was inspired by the stochasticity of brain signals and by the unconventional noise-based communication schemes, such as the Kish cypher.
35.1 The noise-based logic space and hyperspace
The logic values are represented by multi-dimensional "vectors"(orthogonal functions) and their superposition, where the orthogonal basis vectors are independent noises. By the proper combination (products or set-theoretical prod- ucts) of basis-noises, which are called noise-bit, a logic hyperspace can be constructed with D(N) = 2N number of dimensions, where N is the number of noise-bits. Thus N noise-bits in a single wire correspond to a system of 2N N classical bits that can express 22 different logic values. Independent realizations of a stochastic process of zero mean have zero cross-correlation with each other and with other stochastic processes of zero mean. Thus the basis noise vectors are orthogonal not only to each other but they and all the noise-based logic states (superpositions) are orthogonal also to any background noises in the hardware. Therefore, the noise-based logic concept is robust against background noises, which is a property that can potentially offer a high energy-efficiency.
35.2 The types of signals used in noise-based logic
In the paper,[3] where noise-based logic was first introduced, generic stochastic-processes with zero mean were pro- posed and a system of orthogonal sinusoidal signals were also proposed as a deterministic-signal version of the logic system. The mathematical analysis about statistical errors and signal energy was limited to the cases of Gaussian noises and superpositions as logic signals in the basic logic space and their products and superpositions of their prod- ucts in the logic hyperspace (see also.[4] In the subsequent brain logic scheme,[5] the logic signals were (similarly to neural signals) unipolar spike sequences generated by a Poisson process, and set-theoretical unifications (super- positions) and intersections (products) of different spike sequences. Later, in the instantaneous noise-based logic schemes[6][7] and computation works,[8] random telegraph waves (periodic time, bipolar, with fixed absolute value of amplitude) were also utilized as one of the simplest stochastic processes available for NBL. With choosing unit amplitude and symmetric probabilities, the resulting random-telegraph wave has 0.5 probability to be in the +1 or in the −1 state which is held over the whole clock period.
35.3 The noise-based logic gates
Noise-based logic gates can be classified according to the method the input identifies the logic value at the input. The first gates[3][4] analyzed the statistical correlations between the input signal and the reference noises. The advantage of these is the robustness against background noise. The disadvantage is the slow speed and higher hardware complexity.
99 100 CHAPTER 35. NOISE-BASED LOGIC
The instantaneous logic gates[5][6][7] are fast, they have low complexity but they are not robust against background noises. With either neural spike type signals or with bipolar random-telegraph waves of unity absolute amplitude, and randomness only in the sign of the amplitude offer very simple instantaneous logic gates. Then linear or analog devices unnecessary and the scheme can operate in the digital domain. However, whenever instantaneous logic must be interfaced with classical logic schemes, the interface must use correlator-based logic gates for an error-free signal.[6]
35.4 Universality of noise-based logic
All the noise-based logic schemes listed above have been proven universal.[3][6][7] The papers typically produce the NOT and the AND gates to prove universality, because having both of them is a satisfactory condition for the uni- versality of a Boolean logic.
35.5 Computation by noise-based logic
The string verification work[8] over a slow communication channel shows a powerful computing application where the methods is inherently based on calculating the hash function. The scheme is based on random telegraph waves and it is mentioned in the paper[8] that the authors intuitively conclude that the intelligence of the brain is using similar operations to make a reasonably good decision based on a limited amount of information. It is also interesting to note that the superposition of the first D(N) = 2N integer numbers can be produced with only 2N operations, which the authors call “Achilles ankle operation” in the paper.[4]
35.6 Computer chip realization of noise-based logic
Preliminary schemes have already been published[8] to utilize noise-based logic in practical computers. However, it is obvious from these papers that this young field has yet a long way to go before it will be seen in everyday applications.
35.7 References
[1] David Boothroyd (22 February 2011). “Cover Story: What’s this noise all about?". New Electronics.
[2] Justin Mullins (7 October 2010). “Breaking the Noise Barrier: Enter the phonon computer”. New Scientist.
[3] Laszlo B. Kish (2009). “Noise-based logic: Binary, multi-valued, or fuzzy, with optional superposition of logic states”. Physics Letters A 373 (10): 911–918. arXiv:0808.3162. doi:10.1016/j.physleta.2008.12.068.
[4] Laszlo B. Kish; Sunil Khatri; Swaminathan Sethuraman (2009). “Noise-based logic hyperspace with the superposition of 2^N states in a single wire”. Physics Letters A 373 (22): 1928–1934. arXiv:0901.3947. doi:10.1016/j.physleta.2009.03.059.
[5] Sergey M. Bezrukov; Laszlo B. Kish (2009). “Deterministic multivalued logic scheme for information processing and routing in the brain”. Physics Letters A 373 (27–28): 2338–2342. arXiv:0902.2033. doi:10.1016/j.physleta.2009.04.073.
[6] Laszlo B. Kish; Sunil Khatri; Ferdinand Peper (2010). “Instantaneous noise-based logic”. Fluctuation and Noise Letters 09 (4): 323–330. arXiv:1004.2652. doi:10.1142/S0219477510000253.
[7] Peper, Ferdinand; Kish, Laszlo B. (2011). “Instantaneous, Non-Squeezed, Noise-Based Logic”. Fluctuation and Noise Letters 10 (2): 231. doi:10.1142/S0219477511000521.
[8] Laszlo B. Kish; Sunil Khatri; Tamas Horvath (2010). “Computation using Noise-based Logic: Efficient String Verification over a Slow Communication Channel”. The European Physical Journal B 79: 85–90. arXiv:1005.1560. doi:10.1140/epjb/e2010- 10399-x.
35.8 External links
• Homepage of noise-based logic at Texas A&M University Chapter 36
SQLf
SQLf is a SQL extended with fuzzy set theory application for expressing flexible (fuzzy) queries to Regular Relational Databases. Between the known extensions proposed to SQL, at the present time, this is the most complete, because it allows the use of diverse fuzzy elements in all the constructions of the language SQL.[1][2] SQLf is the only known proposal of flexible query system allowing linguistic quantification over set of rows in queries throw the extension of SQL nesting and partitioning structures with fuzzy quantifiers. It also allows the use of quanti- fiers to qualify the quantity of search criteria satisfied by single rows. For query evaluation, they have intended several mechanisms.[3] The more important is the one based on the derivation principle [4] that consists in deriving classic queries that produce, given a threshold t, the t-cut of the result of the fuzzy query, so that the additional processing cost of using a fuzzy language is diminished.
36.1 Basic Block
The fundamental querying structure of SQLf is the multi-relational block. The conception of this structure is based on the three basic operations of the Relational Algebra: Projection, Cartesian Product and Selection, and the application of fuzzy sets’ concepts. The result of a SQLf query is a fuzzy set of rows that is a fuzzy relation instead of a regular relation. A basic block in SQLf consists of a SELECT clause, a FROM clause and a WHERE clause, that is optional. The semantic of this query structure is: The SELECT clause corresponds to the projection. It specifies the relations’ attributes (or attribute expressions) that will be selected. The resulting table is a fuzzy set and the resulting table is in decreasing ordered of satisfaction degree. For shake of simplicity in presentation of query semantic we will assume, without loss of generality, single attribute in SELECT clause. The SELECT clause specifies also a calibration that is intended to restrict the set of rows retrieved. There are two kinds of calibrations: the quantitative and the qualitative. In quantitative calibration the user specifies the number of answer to be retrieved. The query is intended to retrieve the rows with highest membership degrees up to the number of required answers. In qualitative calibration the user specifies a minim level of satisfaction that must have any retrieved row. The FROM clause corresponds to the Cartesian Product. The consult is made on the Cartesian Product of the relations that are specified in this clause. For shake of simplicity in presentation of query semantic we will assume, without loss of generality, single relation FROM clause. The WHERE clause corresponds to the selection. It specifies the condition for which the satisfaction degree will be calculated. Rows that do not satisfy at all the condition are rejected. This condition is a fuzzy predicate that may involve any attribute of the relations. The following is an example of a SELECT query that returns a list of hotels that are cheap. The query retrieves all rows from the Hotels table that satisfice the fuzzy predicate cheap defined by the fuzzy set μ=(∞, ∞, 25, 30). The result is sorted in descending order by the membership degree of the query. The asterisk (*) in the select list indicates that all columns of the Hotels table should be included in the result set. SELECT * FROM Hotels WHERE price = cheap;
101 102 CHAPTER 36. SQLF
36.2 References
[1] Bosc, P.; Pivert, O. (1995). “SQLf: a relational database language for fuzzy querying”. IEEE Transactions on Fuzzy Systems 3 (1): 1–17. doi:10.1109/91.366566. ISSN 1063-6706.
[2] Bosc, P.; Piver, O. (2000). Knowledge Management in Fuzzy Databases. Heidelberg: Physica-Verlag HD. pp. 171–190. ISBN 978-3-7908-1865-9.
[3] Bosc, P.; Pivert, O. (2000). “SQLf Query Functionality on Top of a Regular Relational Database Management System”. pp. 171–190. doi:10.1007/978-3-7908-1865-9_11.
[4] Bosc, P.; Pivert, O. (1995). “On the efficiency of the alpha-cut distribution method to evaluate simple fuzzy relational queries”. World Scientific Publishing: 251–260. Chapter 37
Łukasiewicz logic
In mathematics, Łukasiewicz logic (/luːkəˈʃɛvɪtʃ/; Polish pronunciation: [wukaˈɕɛvʲitʂ]) is a non-classical, many valued logic. It was originally defined in the early 20th-century by Jan Łukasiewicz as a three-valued logic;[1] it was later generalized to n-valued (for all finite n) as well as infinitely-many-valued (ℵ0-valued) variants, both propositional [2] and first-order. The ℵ0-valued version was published in 1930 by Łukasiewicz and Alfred Tarski; consequently it is sometimes called the Łukasiewicz-Tarski logic.[3] It belongs to the classes of t-norm fuzzy logics[4] and substructural logics.[5] This article presents the Łukasiewicz[-Tarski] logic in its full generality, i.e. as an infinite-valued logic. For an elementary introduction to the three-valued instantiation Ł3, see three-valued logic.
37.1 Language
The propositional connectives of Łukasiewicz logic are implication → , negation ¬ , equivalence ↔ , weak conjunction ∧ , strong conjunction ⊗ , weak disjunction ∨ , strong disjunction ⊕ , and propositional constants 0 and 1 . The presence of conjunction and disjunction is a common feature of substructural logics without the rule of contraction, to which Łukasiewicz logic belongs.
37.2 Axioms
The original system of axioms for propositional infinite-valued Łukasiewicz logic used implication and negation as the primitive connectives:
A → (B → A)
(A → B) → ((B → C) → (A → C)) ((A → B) → B) → ((B → A) → A) (¬B → ¬A) → (A → B). Propositional infinite-valued Łukasiewicz logic can also be axiomatized by adding the following axioms to the ax- iomatic system of monoidal t-norm logic:
• Divisibility: (A ∧ B) → (A ⊗ (A → B)) • Double negation: ¬¬A → A.
That is, infinite-valued Łukasiewicz logic arises by adding the axiom of double negation to basic t-norm logic BL, or by adding the axiom of divisibility to the logic IMTL. Finite-valued Łukasiewicz logics require additional axioms.
103 104 CHAPTER 37. ŁUKASIEWICZ LOGIC
37.3 Real-valued semantics
Infinite-valued Łukasiewicz logic is a real-valued logic in which sentences from sentential calculus may be assigned a truth value of not only zero or one but also any real number in between (e.g. 0.25). Valuations have a recursive definition where:
• w(θ ◦ ϕ) = F◦(w(θ), w(ϕ)) for a binary connective ◦,
• w(¬θ) = F¬(w(θ)),
• w(0) = 0 and w(1) = 1, and where the definitions of the operations hold as follows:
• Implication: F→(x, y) = min{1, 1 − x + y}
• Equivalence: F↔(x, y) = 1 − |x − y|
• Negation: F¬(x) = 1 − x
• Weak Conjunction: F∧(x, y) = min{x, y}
• Weak Disjunction: F∨(x, y) = max{x, y}
• Strong Conjunction: F⊗(x, y) = max{0, x + y − 1}
• Strong Disjunction: F⊕(x, y) = min{1, x + y}.
The truth function F⊗ of strong conjunction is the Łukasiewicz t-norm and the truth function F⊕ of strong disjunction is its dual t-conorm. The truth function F→ is the residuum of the Łukasiewicz t-norm. All truth functions of the basic connectives are continuous. By definition, a formula is a tautology of infinite-valued Łukasiewicz logic if it evaluates to 1 under any valuation of propositional variables by real numbers in the interval [0, 1].
37.4 Finite-valued and countable-valued semantics
Using exactly the same valuation formulas as for real-valued semantics Łukasiewicz (1922) also defined (up to iso- morphism) semantics over
• any finite set of cardinality n ≥ 2 by choosing the domain as { 0, 1/(n − 1), 2/(n − 1), ..., 1 }
• any countable set by choosing the domain as { p/q | 0 ≤ p ≤ q where p is a non-negative integer and q is a positive integer }.
37.5 General algebraic semantics
The standard real-valued semantics determined by the Łukasiewicz t-norm is not the only possible semantics of Łukasiewicz logic. General algebraic semantics of propositional infinite-valued Łukasiewicz logic is formed by the class of all MV-algebras. The standard real-valued semantics is a special MV-algebra, called the standard MV- algebra. Like other t-norm fuzzy logics, propositional infinite-valued Łukasiewicz logic enjoys completeness with respect to the class of all algebras for which the logic is sound (that is, MV-algebras) as well as with respect to only linear ones. This is expressed by the general, linear, and standard completeness theorems:[4]
The following conditions are equivalent: 37.6. REFERENCES 105
• A is provable in propositional infinite-valued Łukasiewicz logic • A is valid in all MV-algebras (general completeness) • A is valid in all linearly ordered MV-algebras (linear completeness) • A is valid in the standard MV-algebra (standard completeness).
Font, Rodriguez and Torrens introduced in 1984 the Wajsberg algebra as an alternative model for the infinite-valued Łukasiewicz logic.[6] A 1940s attempt by Grigore Moisil to provide algebraic semantics for the n-valued Łukasiewicz logic by means of his Łukasiewicz–Moisil (LM) algebra (which Moisil called Łukasiewicz algebras) turned out to be an incorrect model for n ≥ 5. This issue was made public by Alan Rose in 1956. C. C. Chang's MV-algebra, which is a model for the ℵ0-valued (infinitely-many-valued) Łukasiewicz-Tarski logic, was published in 1958. For the axiomatically more complicated (finite) n-valued Łukasiewicz logics, suitable algebras were published in 1977 by Revaz Grigolia and called MVn-algebras.[7] MVn-algebras are a subclass of LMn-algebras, and the inclusion is strict for n ≥ 5.[8] In 1982 Roberto Cignoli published some additional constraints that added to LMn-algebras produce proper models for n-valued Łukasiewicz logic; Cignoli called his discovery proper Łukasiewicz algebras.[9]
37.6 References
[1] Łukasiewicz J., 1920, O logice trójwartościowej (in Polish). Ruch filozoficzny 5:170–171. English translation: On three- valued logic, in L. Borkowski (ed.), Selected works by Jan Łukasiewicz, North–Holland, Amsterdam, 1970, pp. 87–88. ISBN 0-7204-2252-3
[2] Hay, L.S., 1963, Axiomatization of the infinite-valued predicate calculus. Journal of Symbolic Logic 28:77–86.
[3] Lavinia Corina Ciungu (2013). Non-commutative Multiple-Valued Logic Algebras. Springer. p. vii. ISBN 978-3-319- 01589-7. citing Łukasiewicz, J., Tarski, A.: Untersuchungen über den Aussagenkalkül. Comp. Rend. Soc. Sci. et Lettres Varsovie Cl. III 23, 30–50 (1930).
[4] Hájek P., 1998, Metamathematics of Fuzzy Logic. Dordrecht: Kluwer.
[5] Ono, H., 2003, “Substructural logics and residuated lattices — an introduction”. In F.V. Hendricks, J. Malinowski (eds.): Trends in Logic: 50 Years of Studia Logica, Trends in Logic 20: 177–212.
[6] http://journal.univagora.ro/download/pdf/28.pdf citing J. M. Font, A. J. Rodriguez, A. Torrens, Wajsberg Algebras, Stochas- tica, VIII, 1, 5-31, 1984
[7] Lavinia Corina Ciungu (2013). Non-commutative Multiple-Valued Logic Algebras. Springer. pp. vii–viii. ISBN 978-3- 319-01589-7. citing Grigolia, R.S.: “Algebraic analysis of Lukasiewicz-Tarski’s n-valued logical systems”. In: Wójcicki, R., Malinkowski, G. (eds.) Selected Papers on Lukasiewicz Sentential Calculi, pp. 81–92. Polish Academy of Sciences, Wroclav (1977)
[8] Iorgulescu, A.: Connections between MVn-algebras and n-valued Łukasiewicz–Moisil algebras—I. Discrete Math. 181, 155–177 (1998) doi:10.1016/S0012-365X(97)00052-6
[9] R. Cignoli, Proper n-Valued Łukasiewicz Algebras as S-Algebras of Łukasiewicz n-Valued Propositional Calculi, Studia Logica, 41, 1982, 3-16, doi:10.1007/BF00373490
37.7 Further reading
• Rose, A.: 1956, Formalisation du Calcul Propositionnel Implicatif ℵ0 Valeurs de Łukasiewicz, C. R. Acad. Sci. Paris 243, 1183–1185.
• Rose, A.: 1978, Formalisations of Further ℵ0-Valued Łukasiewicz Propositional Calculi, Journal of Symbolic Logic 43(2), 207–210. doi:10.2307/2272818
• Cignoli, R., “The algebras of Lukasiewicz many-valued logic - A historical overview,” in S. Aguzzoli et al.(Eds.), Algebraic and Proof-theoretic Aspects of Non-classical Logics, LNAI 4460, Springer, 2007, 69- 83. doi:10.1007/978-3-540-75939-3_5 106 CHAPTER 37. ŁUKASIEWICZ LOGIC
37.8 Text and image sources, contributors, and licenses
37.8.1 Text
• Adaptive neuro fuzzy inference system Source: https://en.wikipedia.org/wiki/Adaptive_neuro_fuzzy_inference_system?oldid=644433950 Contributors: Glenn, Klemen Kocjancic, Rjwilmsi, Tony1, SmackBot, Mwtoews, Andyjsmith, Михајло Анђелковић, Magioladitis, Katharineamy, Casablanca2000in, Wilhelmina Will, SchreiberBike, Boleyn, Addbot, CactusWriter, Yobot, AnomieBOT, WikitanvirBot, Njsg, ZéroBot, Erpankajj, Charon77, BattyBot and Anonymous: 8 • Bate’s chip Source: https://en.wikipedia.org/wiki/Bate’{}s_chip?oldid=515159264 Contributors: Rjwilmsi, Smallman12q, DASHBot and Anonymous: 1 • BL (logic) Source: https://en.wikipedia.org/wiki/BL_(logic)?oldid=606007476 Contributors: EmilJ, Rjwilmsi, Cnoguera, BertSeghers, SiddMahen and Anonymous: 2 • Combs method Source: https://en.wikipedia.org/wiki/Combs_method?oldid=651849327 Contributors: Michael Hardy, Furrykef, Tag- ishsimon, JohnyDog, SuperDude115, Julian Mendez, Burzmali, Melcombe, Lightbot and Anonymous: 5 • Construction of t-norms Source: https://en.wikipedia.org/wiki/Construction_of_t-norms?oldid=610239781 Contributors: Michael Hardy, Woohookitty, Joe Decker, David Eppstein, STBot, LBehounek, GirasoleDE, Toddst1, Yobot, AnomieBOT, Harold Philby, Ms.Entropy, Barakafrit, Ghevriel and Anonymous: 5 • Defuzzification Source: https://en.wikipedia.org/wiki/Defuzzification?oldid=630154626 Contributors: Furrykef, Themusicgod1, Lor- dAmeth, MarSch, Mathbot, Predictor, SmackBot, CBM, Tulkolahten, Kilmer-san, Wireless friend, Luckas-bot, Yobot, BertSeghers, ZéroBot, ClueBot NG, Sairp and Anonymous: 9 • Degree of truth Source: https://en.wikipedia.org/wiki/Degree_of_truth?oldid=630169465 Contributors: Cherkash, Charles Matthews, Furrykef, Hyacinth, Auric, Paul Murray, Falcon Kirtaran, Stephan Leclercq, Paul August, Vanished user lp09qa86ft, Melaen, Simetrical, Rjwilmsi, Tadanisakari, Chris Capoccia, SmackBot, Hippo43, Dreadstar, Bjankuloski06en~enwiki, Simeon, Gregbard, Cydebot, Rgheck, Helgus, SE16, Addbot, ZéroBot, Nizamibilal1064, Nathanielfirst and Anonymous: 9 • European Society for Fuzzy Logic and Technology Source: https://en.wikipedia.org/wiki/European_Society_for_Fuzzy_Logic_and_ Technology?oldid=669523117 Contributors: Wavelength, SmackBot, Spiritia, Valepert, Cydebot, Blanchardb, Sun Creator, Eulenreich, BattyBot and ChrisGualtieri • Fuzzy architectural spatial analysis Source: https://en.wikipedia.org/wiki/Fuzzy_architectural_spatial_analysis?oldid=587639343 Con- tributors: Rjwilmsi, David Eppstein, Alokprasad84, RjwilmsiBot, Slawekb, Tanzeelahad, BG19bot, Xca777, ChrisGualtieri and Anony- mous: 3 • Fuzzy associative matrix Source: https://en.wikipedia.org/wiki/Fuzzy_associative_matrix?oldid=637829355 Contributors: Furrykef, Giftlite, Robofish, Svick, Cbyronpowell, ClueBot, Bgeelhoed, Skyerise and Anonymous: 7 • Fuzzy classification Source: https://en.wikipedia.org/wiki/Fuzzy_classification?oldid=664046587 Contributors: Brianhe, Reyk, Smack- Bot, Gregbard, Joshua Issac, Biggerj1, Mild Bill Hiccup, MatthewVanitas, PJ Geest, Michaelkaufmann, BG19bot, ChrisGualtieri and Anonymous: 2 • Fuzzy cognitive map Source: https://en.wikipedia.org/wiki/Fuzzy_cognitive_map?oldid=660714430 Contributors: Mdd, Rjwilmsi, Tech- notaoist, Megadix~enwiki, Kc2idf, Kuru, Iridescent, Widefox, Magioladitis, Martynas Patasius, Fusion7, Jackaranga, Rodtaber, CBRichter, Adrian Portsmouth, SpellingBot, Yobot, AnomieBOT, FrescoBot, Michael bachhofer, Mglikas, Helpful Pixie Bot, BG19bot, Tom Pip- pens, Dianeblack, EvergreenFir, Tech-Manager and Anonymous: 17 • Fuzzy Control Language Source: https://en.wikipedia.org/wiki/Fuzzy_Control_Language?oldid=583232981 Contributors: Furrykef, Burn, Allen3, Marudubshinki, Ffangs, SmackBot, JustAnotherJoe, Bairam, Meiskam, Addbot, Mur en and Anonymous: 6 • Fuzzy control system Source: https://en.wikipedia.org/wiki/Fuzzy_control_system?oldid=663567927 Contributors: The Anome, Ap, Aldie, Stevertigo, Kku, MartinHarper, Wapcaplet, Ronz, CatherineMunro, Nikai, Furrykef, Boffy b, Naddy, Gandalf61, Rdsmith4, Sam Hocevar, Neutrality, Dave Foley, Kaszeta, Nickj, Viriditas, R. S. Shaw, Mdd, Alex.g, Firsfron, BD2412, Grammarbot, Penumbra2000, DVdm, AntoineHersen, YurikBot, Gaius Cornelius, SAE1962, Thiseye, EverettColdwell, Closedmouth, Arthur Rubin, Reyk, SmackBot, Slashme, Oli Filth, EncMstr, JustAnotherJoe, SpiderJon, Acdx, Ryulong, Passino, Gregbard, Pcarew, Wikid77, Gerla314, Destynova, Aon~enwiki, Oosterwal, CommonsDelinker, Erkan Yilmaz, Maurice Carbonaro, Gerla, Guillaume2303, Spinningspark, Prakash Nad- karni, ImageRemovalBot, Dlrohrer2003, ClueBot, Snigbrook, PixelBot, SchreiberBike, XLinkBot, Dthomsen8, NellieBly, Addbot, Ya- sunat, MrOllie, Luckas-bot, Yobot, AnomieBOT, The Firewall, JanJantzen, Ws no1, T2gurut2, Drwu82, Boxplot, RedBot, Puzl bustr, EmausBot, Dzkd, Tijfo098, ClueBot NG, Widr, Yncn, Helpful Pixie Bot, HMSSolent, Bmusician, Kollamrajeshr, Northamerica1000, Paloma01, Bob Wont Die, VmayaV, Adamsmall22, Ashwinigoud, Royzeng, Loslix, Wikiwizkidd, Mordecai higgenbotham, Boldscience, Brzydalski, Paheld, Mje123 and Anonymous: 73 • Fuzzy electronics Source: https://en.wikipedia.org/wiki/Fuzzy_electronics?oldid=544031107 Contributors: Heron, Edward Z. Yang, Timl, Atlant, Toffile, Alynna Kasmira, Jpbowen, SmackBot, Lindosland, Bjankuloski06en~enwiki, Amalas, PamD, KoenDelaere, Clue- Bot, Licknuts, Addbot, Xqbot, EricWesBrown and Anonymous: 6 • Fuzzy finite element Source: https://en.wikipedia.org/wiki/Fuzzy_finite_element?oldid=630169477 Contributors: Hyacinth, Bender235, Reyk, SmackBot, Gregbard, EagleFan, KoenDelaere, Addbot, Yobot, Persian knight shiraz, ZéroBot and Frietjes • Fuzzy logic Source: https://en.wikipedia.org/wiki/Fuzzy_logic?oldid=670685809 Contributors: Damian Yerrick, Tarquin, Ap, Rjstott, Christian List, Heron, Stevertigo, RTC, Michael Hardy, Pit~enwiki, Ixfd64, Eric119, Ahoerstemeier, Ronz, Harry Wood, AugPi, An- dres, Palfrey, EdH, Loren Rosen, Zoicon5, Markhurd, Furrykef, Hyacinth, Omegatron, Traroth, Robbot, Academic Challenger, Rursus, Blainster, Ruakh, Tobias Bergemann, Cedars, Giftlite, Zaphod Beeblebrox, Duniyadnd, Jason Quinn, Gyrofrog, Lawrennd, Quackor, Marcus Beyer, L353a1, Gauss, Icairns, Zfr, TreyHarris, Ohka-, Clemwang, Kadambarid, Xezbeth, Mani1, Paul August, Guard, El- wikipedista~enwiki, Mr. Billion, El C, Chalst, Moilleadóir, Causa sui, Smalljim, R. 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Clarke, Sqzx, Drift chambers, Sn1per, M.r.ebraahimi, WikiHannibal, Colbert Sesanker, Xca777, Flaminchimp, Diglio.simoni, ShashankSharma2511, Barakafrit, Illia Connell, Керен, Aklnih, Suraduttashandilya, Jochen Burghardt, Funnyperson22, Phmresearch, , Eknigge, Pdecalculus, Jumpulse, Zsof- ,Renates45, Dexalkaline, Sairp ,בשלני ,tua, Maple2013, Julaei, RudiSeising, Wangbo66653, Jptvgrey, Bilorv, Monkbot, Gregusmihai Mrityunjaykr02, TranquilHope, Qzekrom, William Zachary Runyon, Brewstoo, Analplays, Aangell123, Mcconnellsc58, Sigma.4292, SocraticOath, Charlottecourtleeds and Anonymous: 451 • Fuzzy markup language Source: https://en.wikipedia.org/wiki/Fuzzy_markup_language?oldid=661029091 Contributors: Bearcat, Red- Wolf, Malcolma, Hahc21, Yobot, Mjs1991, RjwilmsiBot, SporkBot, Doctree, Bmusician, Northamerica1000, Gioaca, Lia84, Monkbot and Anonymous: 4 • Fuzzy mathematics Source: https://en.wikipedia.org/wiki/Fuzzy_mathematics?oldid=625455159 Contributors: Michael Hardy, Mun- ford, Orlady, Uffish, Wavelength, SmackBot, Jagged 85, Martijn Hoekstra, Plvekamp, R'n'B, JohnBlackburne, LBehounek, AgentCDE, Shaliya waya, Batyr, LemmaDance, Addbot, Yobot, Crystal whacker, Mishka.medvezhonok, Cat 1012000, BertSeghers, Nkf31, Helpful Pixie Bot, Brad7777, AK456, Magnolia677, RudiSeising, Rgcase, Debdas.email and Anonymous: 13 • Fuzzy measure theory Source: https://en.wikipedia.org/wiki/Fuzzy_measure_theory?oldid=604820507 Contributors: Michael Hardy, Jason Quinn, Mdd, Oleg Alexandrov, YurikBot, Srinivasasha, Hirak 99, SmackBot, Diegotorquemada, Bluebot, Bejnar, CBM, Greg- bard, Alaibot, Epbr123, Helgus, JCarlos, JohnBlackburne, LBehounek, VanishedUserABC, Dmcq, Jojalozzo, Melcombe, Kajuna70, LevinCoolXYZ, Cp111, Addbot, DOI bot, Anonash, Datandrews, Citation bot 1, Slimey J, RjwilmsiBot, Brad7777, ChrisGualtieri, Mogism, Mark viking and Anonymous: 18 • Fuzzy number Source: https://en.wikipedia.org/wiki/Fuzzy_number?oldid=666520313 Contributors: AugPi, Nikkimaria, Yamaguchi , Cybercobra, Bjankuloski06en~enwiki, De2k, David Eppstein, KoenDelaere, Curtdbz, Excirial, Addbot, Mrocklin, Luckas-bot, Dude1818, MusikAnimal, AvocatoBot and Anonymous: 7 • Fuzzy pay-off method for real option valuation Source: https://en.wikipedia.org/wiki/Fuzzy_pay-off_method_for_real_option_valuation? oldid=527539777 Contributors: Michael Hardy, Fintor, Lmatt, Malcolma, Ulner, Dancter, Mild Bill Hiccup, Yobot, Mikc75, Erik9bot, Hmainsbot1 and Anonymous: 8 • Fuzzy routing Source: https://en.wikipedia.org/wiki/Fuzzy_routing?oldid=567513385 Contributors: The Anome, Doco, Sceptre, Sand- stein, SmackBot, Kvng, CmdrObot, Mercury~enwiki, Dawnseeker2000, Zxiiro, Thanhantorob, 1ForTheMoney, ChrisGualtieri and Lesser Cartographies • Fuzzy rule Source: https://en.wikipedia.org/wiki/Fuzzy_rule?oldid=630169471 Contributors: Hyacinth, Andreas Kaufmann, Tinctorius, Bjankuloski06en~enwiki, Gregbard, Alaibot, Dikisoccer, Addbot, MrOllie, Computationalverb and Anonymous: 7 • Fuzzy set Source: https://en.wikipedia.org/wiki/Fuzzy_set?oldid=669262232 Contributors: Zundark, Taw, Toby Bartels, Boleslav Bob- cik, Michael Hardy, MartinHarper, Ixfd64, Tgeorgescu, Александър, AugPi, Palfrey, Evercat, Charles Matthews, Markhurd, Furrykef, Hyacinth, Grendelkhan, VeryVerily, Robbot, Jaredwf, Peak, Giftlite, Jcobb, Duncharris, Jason Quinn, Phe, Urhixidur, Elwikipedista~enwiki, El C, Kwamikagami, R. S. Shaw, Pinar, Kusma, Joriki, Smmurphy, Ryan Reich, Salix alba, Mathbot, Predictor, YurikBot, Wavelength, Michael Slone, SpuriousQ, Gaius Cornelius, Srinivasasha, Supten, Jurriaan, Ml720834~enwiki, SmackBot, Hydrogen Iodide, Commander Keane bot, Dreadstar, Rijkbenik, Bjankuloski06en~enwiki, Valepert, Elharo, JRSpriggs, George100, Paulmlieberman, CRGreathouse, Ksoileau, Gregbard, VashiDonsk, NotQuiteEXPComplete, Helgus, Nick Number, Abdel Hameed Nawar, Михајло Анђелковић, MER- C, Ty580, Bouktin, Magioladitis, MartinBot, Maurice Carbonaro, Gerla, DoorsAjar, Krzysiulek~enwiki, BotKung, LBehounek, In- formationSpace, Kilmer-san, VanishedUserABC, Cesarpermanente, ClueBot, Lukipuk, QYV, Pgallert, Multipundit, Addbot, Wireless friend, Legobot, Yobot, AnomieBOT, DemocraticLuntz, Riyad parvez, Pownuk, J JMesserly, Charvest, T2gurut2, Kierkkadon, Tin- ton5, Carel.jonkhout, FoxBot, Mjs1991, DixonDBot, The tree stump, WikitanvirBot, Matsievsky, Tijfo098, ChuispastonBot, ClueBot NG, Dezireh batist, Frietjes, Helpful Pixie Bot, StarryGrandma, Zbhsueh, Dannyeuu, Jcallega, Mark viking, Faizan, DangerouslyPersua- siveWriter, Atharkharal, IITHemant, Reddraggone9, RudiSeising, JMP EAX, Ffswontforget3 and Anonymous: 92 • Fuzzy set operations Source: https://en.wikipedia.org/wiki/Fuzzy_set_operations?oldid=646750851 Contributors: Michael Hardy, Charles Matthews, Hyacinth, Jason Quinn, Quietly, Pearle, Mailer diablo, Woohookitty, Rococo roboto, Mathbot, Predictor, SmackBot, Pep- sidrinka, Tanber, Jon Awbrey, Konerak, Beetstra, Gregbard, WinBot, R'n'B, LBehounek, Drwu82, Erik9bot, Cannolis, GrayFullbuster, J.Paskalis and Anonymous: 19 108 CHAPTER 37. ŁUKASIEWICZ LOGIC
• Fuzzy Sets and Systems Source: https://en.wikipedia.org/wiki/Fuzzy_Sets_and_Systems?oldid=646021975 Contributors: George100, Alastair Haines, T@nn, DGG, Guillaume2303, LBehounek, GirasoleDE, MatthewVanitas, Fgnievinski, Abductive, Kajervi, J.vanderboom, ArmbrustBot and Anonymous: 1 • Fuzzy subalgebra Source: https://en.wikipedia.org/wiki/Fuzzy_subalgebra?oldid=503697017 Contributors: Mdd, Rjwilmsi, Chrispounds, SmackBot, Alaibot, Gerla, LokiClock, ChrisGualtieri and Anonymous: 4 • Fuzzy transportation Source: https://en.wikipedia.org/wiki/Fuzzy_transportation?oldid=601791032 Contributors: Michael Hardy, Nawl- inWiki, Sarah, David Eppstein, KathrynLybarger, Random Fixer Of Things, Ghatee, Solidkuzma, Yobot, AnomieBOT, Abductive, Jesse V., GoingBatty and Anonymous: 2 • High Performance Fuzzy Computing Source: https://en.wikipedia.org/wiki/High_Performance_Fuzzy_Computing?oldid=572435736 Contributors: Malcolma, R'n'B, Addbot, Yobot, FrescoBot, Vrenator, SporkBot, EnKukoc and Anonymous: 1 • Linear partial information Source: https://en.wikipedia.org/wiki/Linear_partial_information?oldid=622335026 Contributors: Michael Hardy, Alan Liefting, Tagishsimon, Andreas Kaufmann, BD2412, Deville, DMS, CmdrObot, Ntsimp, Thijs!bot, Destynova, VolkovBot, Krzysiulek~enwiki, Melcombe, Foxj, SchreiberBike, BOTarate, Addbot, Lightbot, Obersachsebot, Gamewizard71, Frietjes and Anony- mous: 1 • Membership function (mathematics) Source: https://en.wikipedia.org/wiki/Membership_function_(mathematics)?oldid=661980255 Contributors: Charles Matthews, Taxman, Ukexpat, R. S. Shaw, Smmurphy, Salix alba, Srinivasasha, Samuel Blanning, Bjankuloski06en~enwiki, Iridescent, Helgus, Hvtuananh, R'n'B, LBehounek, Addbot, Xqbot, Doh5678, Shashanksingh.1102, Nadadur and Anonymous: 5 • Monoidal t-norm logic Source: https://en.wikipedia.org/wiki/Monoidal_t-norm_logic?oldid=607377546 Contributors: Edward, EmilJ, Rjwilmsi, Trovatore, CWenger, Iridescent, Gregbard, Thenub314, LBehounek, Yobot, Omnipaedista and Anonymous: 4 • MV-algebra Source: https://en.wikipedia.org/wiki/MV-algebra?oldid=622291636 Contributors: Michael Hardy, GTBacchus, Charles Matthews, Eequor, EmilJ, Kjkolb, Dismas, Woohookitty, Ruud Koot, Smmurphy, Salix alba, Wavelength, Riverofdreams, Mhss, Greg- bard, Julian Mendez, LBehounek, Ululuca, Hans Adler, Pgallert, Addbot, Yobot, AnomieBOT, Charvest, Tijfo098, Victorpablosceruelo, Mark viking, JMP EAX and Anonymous: 9 • Noise-based logic Source: https://en.wikipedia.org/wiki/Noise-based_logic?oldid=596653440 Contributors: Headbomb, SchreiberBike, MatthewVanitas, Citation bot, FrescoBot, Tom.Reding, Wcherowi, Repep and Anonymous: 4 • SQLf Source: https://en.wikipedia.org/wiki/SQLf?oldid=668356763 Contributors: Michael Hardy, Bgwhite, Yobot, I dream of horses and Gssbzn • Łukasiewicz logic Source: https://en.wikipedia.org/wiki/%C5%81ukasiewicz_logic?oldid=665733591 Contributors: Michael Hardy, Hyacinth, Giftlite, Kwamikagami, EmilJ, Gene Nygaard, Oleg Alexandrov, Ruud Koot, Reetep, Ritchy, Arthur Rubin, Donhalcon, Smack- Bot, Mhss, CBM, Deflective, LBehounek, Soler97, Francvs, Luckas-bot, AnomieBOT, RjwilmsiBot, Tijfo098, Doh5678, KLBot2, JMP EAX and Anonymous: 10
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